the gravity field of venus and its relationship to the...

The Gravity Field of Venus and Its Relationship tothe Dynamic Processes in the Mantle

Research report to the Czech National Grant No. 205/02/1306∗

Martin Pauer

∗ This report was submitted as a diploma thesisat the Department of Geophysics, Faculty of Mathematics and Physics,

Charles University, Prague, in April 2004.

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Here I would like to give thanks to all the people who helped me in writing this diplomathesis. First of all to Ondrej Cadek for his kind care and to Jirı Zahradnık for his everlastingenthusiasm. Then also to Ludek Vecsey for his help in investigating wavelet analysis and toHana Cızkova and Ctirad Matyska for their helpful advice and discussions. To the rest ofour department for their kind and personal attitude toward the students, to Ondrej Santolıkfor his nice lectures on the magnetosphere, to Marie Behounkova for her valuable supportduring my studies, to Casey Heagerty for proofreading this thesis and to Ladislav Hanykfor his genuinely good mood.

Many thanks belong to my whole family who have supported me all these years and tomy friends – without them I would probably not have completed my studies. And of courseto my dear Martina who believed in me more than I did ...

Because the planetary sciences are not yet the subject of general studies in Czech I haveseveral times during my work on this thesis asked foreign academic colleagues for help ormaterials, which they kindly provided. Therefore I wish to thank also Gerald Schubert(UCLA), Tilman Spohn (DLR), Ulrich Christensen (Max Planck Institute fur Aeronomie),David Yuen (MSI), Greg Neumann (MIT), Alex Konopliv (JPL) and Donald Gurnett (Uni-versity of Iowa).

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Contents

1 Introduction 9

2 Venus As We Know It 112.1 Structure of Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.4 Crust, Lithosphere and Mantle . . . . . . . . . . . . . . . . . . . . . 212.1.5 Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Evolution of Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 Resurfacing Event Constraint . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Heat Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.3 Convection Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Quest for Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Comparison with Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Data Sets 273.1 Spherical Harmonic Models for Venus . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 Geoid model MGNP180U . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 Topography model GTDR.3 . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Spherical Harmonic Models for Other Planets . . . . . . . . . . . . . . . . . 31

4 Isostasy and Crustal Structure 374.1 Uncompensated Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Isostatic Compensation of Topography . . . . . . . . . . . . . . . . . . . . . 384.3 Degree ADC Method for Other Planets . . . . . . . . . . . . . . . . . . . . . 42

5 Dynamic Model of Geoid 45

6 Localization Analysis 536.1 Localization of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Localization of Correlation and Admittance . . . . . . . . . . . . . . . . . . 586.3 Emax-kmax Method Applied on Venus’ Geoid . . . . . . . . . . . . . . . . . . 586.4 Emax-kmax Analysis for Other Planets . . . . . . . . . . . . . . . . . . . . . . 61

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CONTENTS 4

7 Discussion of Results 71

8 Conclusions 73

A Spherical Harmonics 75A.1 Scalar Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76A.2 Vector Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 80A.3 Tensor Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.4 Operations with Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . 82

B Spectral-spatial Methods 89B.1 Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . 89

B.1.1 1D Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90B.1.2 2D Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

B.2 2D Spherical Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91B.2.1 Spectral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B.2.2 Spatial Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

B.3 Compositional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

C Dynamic Modelling 95C.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95C.2 Parametrization and Numerical Solution . . . . . . . . . . . . . . . . . . . . 96C.3 Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

References 103

Internet References 109

Nazev prace: Gravitacnı pole Venuse a jeho vztah k dynamickym procesum v jejım plastiAutor: Martin PauerKatedra (ustav): Katedra geofyziky, MFF UK v PrazeVedoucı diplomove prace: Doc. RNDr. Ondrej Cadek, CSc.E-mail vedoucıho: [email protected]: Modelovanım mechanismu vzniku geoidu Venuse lze zıskat cenne poznatkyo vnitrnı strukture planety. Spektrum geoidu se nam darı rozdelit na dve casti a kazdouz nich vysvetlit samostatne v ramci jine teorie kompenzace topografie. Na stupnıch j > 40je vhodnym mechanismem izostaticka kompenzace kury o jedne vrstve s mocnostı 35 km; nastupnıch j=2-40 je vhodnym kompenzacnım mechanismem dynamicke tecenı v plasti. Tov teto praci modelujeme za pouzitı teorie odezvovych funkcı. Za predpokladu vztlakovychsil menıcıch se pouze lateralne se nam darı vystihnout kolem 90% pozorovaneho geoidu i to-pografie. Resenım obracene ulohy pro viskoznı strukturu dostavame litosferu o mocnosti100-200 km s viskozitou o nekolik radu vyssı v porovnanı s hlubsımi partiemi. Pro zbytekplaste pak dostavame narust viskozity smerem k jadru o 1-1.5 radu bez prıtomnosti ostreohranicene zony nızke viskozity (podobne astenosfere u Zeme). Predikovane hodnoty dy-namickeho geoidu i topografie jsou analyzovany pomocı spektralnıch i lokalizacnıch metoda vykazujı dobrou korelaci s pozorovanymi daty.Klıcova slova: Venuse, struktura plaste, viskozita, gravitacnı pole

Title: The Gravity Field of Venus and Its Relationship to Dynamic Processes in the MantleAuthor: Martin PauerDepartment: Department of Geophysics, MFF UK in PragueSupervisor: Doc. RNDr. Ondrej Cadek, CSc.Supervisor’s e-mail address: [email protected]: Modelling Venus’ gravity gives us acceptable constraints on the structure of theplanet. We can successfully divide the geoid spectrum into two main parts and each of themexplain on the basis of different topographic support. For a signal at degrees j > 40 it isappropriate to use the crustal-isostasy model with the one-layer crust of thickness 35 km; fordegrees j=2-40 is the suitable compensation mechanism a dynamic flow in the mantle. Inour work this is investigated in the framework of internal loading theory. When we assumethat the buoyancy force is only varying laterally we can explain well about 90% of both thegeoid and topography. By solving the inverse problem for a viscosity structure we obtaina lithosphere with a thickness 100-200 km and of a viscosity stiffer by several orders ofmagnitude than the deeper parts. For the rest of the mantle we obtain a viscosity increasetowards the core by 1-1.5 orders of magnitude with no indication of a narrow low-viscosityzone (similar to Earth’s asthenosphere). The predicted dynamic geoid and topography areanalyzed by means of both spectral and localization methods and show a good correlationto the observed data.Keywords: Venus, mantle structure, viscosity, gravity

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Earth is the cradle of mankind.But one cannot live in the cradle forever.

K. E. Ciolkovski (1857-1935)

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Chapter 1

Introduction

In the case of planets other than Earth our knowledge of their interiors is very poor because,except for the Moon, we have no seismological observations which can give us direct infor-mation about their structure. Our only tool to uncover the subsurface structure and mantleproperties is to carefully analyze the gravity/topography relations – both of these fields areprecisely measured by orbiting space probes. This work’s aim is to continue such effortsin the case of Venus, previously studied by other authors, with modifications of currentlyused methods and employing new data models obtained by recent analyses. Our effort is toidentify the part of the geoid generated by dynamic processes in the mantle and then, bymeans of the inverse problem, find the best rheological model to reproduce it.

Basically this thesis is divided into two parts. Chapter 2 has a character of literatureretrieval, where all the different findings concerning geophysics of Venus are reviewed –for identification of interdisciplinary links this section also contains parts focused on themagnetosphere and atmosphere of Venus. Chapter 3 deals with the used spherical harmonicmodels of geoid and topography and with the techniques of obtaining them.

Our own contribution to the understanding of Venus’ structure comes in the chaptersafter these. In Chapter 4 the global estimates of the crustal thickness based on the isostasyfor short wavelengths are made, and in Chapter 5 we applied the internal loading theoryto explain the dynamic geoid and topography which are believed to be observed at longwavelengths on Venus. The test of our results by the multidimensional methods is done inChapter 6 and the discussion of it with regards to the work of other authors in Chapter 7.

Because some of the materials included in this thesis are obtained from the internet,their reference is noted as [i?] and the list of sources is given at the end of the thesis.The supplemental CD contains the PDF version of this work, the source files of geoid andtopography coefficients and some of the results in EPS format.

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CHAPTER 1. INTRODUCTION 10

Chapter 2

Venus As We Know It

From the very beginnings of civilization people have known Venus as a morning or eveningstar. Its dense clouds (Fig. 2.1) reflect incident sun light so intensely that they could notoverlook it. The Greeks named such stars which wandered through the sky among otherfixed stars ”planets”, which means travellers. And perhaps for its brightness this planet gotits name from the Roman goddess of beauty and love, Venus.

But the same dense clouds prevented the first astronomers from observing its surfaceafter the telescope was invented. However, because of the physical similarity between Venusand our planet, it was for a long time accepted that the surface conditions would be thesame. We had to wait until the radiotelescope was constructed and used for purposes ofastronomy before we could learn more about this planet. The first radio reflections gavepeople suggestions of what could be hidden under the dense venusian atmosphere.

Figure 2.1: Venus observed in visible light [i1]

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CHAPTER 2. VENUS AS WE KNOW IT 12

After WWII the time came to go into the space – first Sputnik in 1957 and then otherspacecrafts. Both Russians and Americans tried to reach Venus with unmanned spaceprobes – russian Veneras several times landed softly on the surface of Venus and workeduntil the hostile environment destroyed them. Americans succeeded with Pioneer Venus andMagellan missions to get into orbit, and so obtained important global observations.

People always speculated about Venus – maybe because it looked so similar to Earth.After getting the first data from space missions, the geoscientists focused on applying theirknowledge of our planet to this body. Nowadays there is whole new scientific disciplinecalled planetology and a lot of people work seriously on Venus research. Many questionshave already been answered but more remain yet unsolved – one of them is the origin ofVenus’ geoid. Answering these questions will maybe lead us to discover of general rules ofplanetary behavior and this could help us to better understand our own planet Earth.

2.1 Structure of Venus

All we know about Venus are data recorded at ground-based facilities placed here on Earth,obtained by orbiting probes or by several landers – no interior data are available to help usif thinking about the structure and processes in Venus’ interior. Therefore we often considerit as Earth’s twin and try to apply the knowledge of our planet to our understanding ofVenus. This assumption is allowed mainly because of the shared conditions during theSolar System’s formation – the material composition does not differ too much, the maindifferences were probably just the abundance of water on Earth and the incoming energyfrom the Sun (which is greater on Venus because it is closer to our star than Earth).

2.1.1 Magnetic Field

Most of the planets of the Solar System have huge magnetospheres. These structures areoften 10-100 times larger than the planet itself and in the cases of Mercury, Earth and thegiant planets originate in the interior of the planets by the dynamo processes.

But in the case of Venus – as for comets – there is no internal mechanism producing themagnetic field. However, it is the interaction of a solar wind with the ionosphere (the part ofthe atmosphere with charged particles) that produces a weak magnetic field (e.g. Phillipsand McComas, 1991). This field shows a behavior similar to an internally-generated oneexcept for the dipole characteristics – the structure is shown in Fig. 2.2.

There the bow shock is formed in front of the ionosphere where the particles of thesolar wind are decelerating to sub-Alfvenic and sub-sonic speeds. The magnetic lines and soeven the solar wind particles accumulate in the magnetosheath behind the bow shock. Thismagnetosheath is separated from the ionosphere by the ionopause – here the plasma pressureis in the equilibrium with thermal pressure of venusian atmosphere. The interplanetarymagnetic field is draped around the planet and forms an induced magnetotail behind itsimilar to Earth’s but with polarity controlled solely by the surrounding field. Without thedipole field there is no system of trapped particles surrounding Venus.


Figure 2.2: Structure of Venus’ induced magnetosphere [i2]

2.1.2 Atmosphere

In the early state of the formation of the Solar System Venus probably had an atmospherevery similar to that of the Earth, both in physical properties and chemical composition.But because of global warming (caused by the greenhouse effect) the water vaporized andpartially escaped from the planet, the atmosphere overheated and has stayed in this statetrough the present.

Chemical element Fraction

Carbon Dioxide CO2 96.5%

Nitrogen N2 3.5%

Sulfur Dioxide SO2 150 ppm

Argon Ar 70 ppm

Water H2O 20 ppm

Carbon Monoxide CO 17 ppm

Helium He 12 ppm

Table 2.1: Chemical composition of Venus’ atmosphere (de Pater and Lissauer, 2001)

The question remains of how much water was there. The measured ratio of deuteriumand hydrogen (D:H) is much higher (∼ 0.016) than on Earth (0.00015) (as shown e.g. byDonahue and Hodges, 1992). That could mean Venus may have once had a certainvolume of surface water (where both D and H were contained) but this evaporated as thegreenhouse effect intensified. Because of the lack of a magnetic field (see previous section)which protects the planet and its atmosphere from the solar-wind’s high energies particles,the light isotopes of hydrogen were expelled to space and the D:H ratio increased. If this isthe case, Venus had at least a few tenths of a percent of the Earth’s hydrosphere.


On the other hand a second interpretation for this D/H measurement indicates thepossibility that Venus’ atmosphere is permanently supplied with hydrogen from some othersource (Grinspoon, 1993). By studying the escape time of atmospheric particles thehistoric reservoir of water is doubted and a permanent outgassing from volcanoes or otherinternal sources is suggested. Possible connections to resurfacing event are also arisingbecause of a supposed fractionation of subducted surface crust. Which of these theories isvalid will be proven by future probes measuring precisely the surface chemical composition.

The atmosphere of Venus is very dense compared to that of Earth – the pressure atthe surface reaches 92 bars and the density is 65 kg·m−3 (on Earth it is only 1.22 kg·m−3).The chemical composition is similar to the Earth’s atmosphere except that the dominantelement is not nitrogen but carbon dioxide (which is the cause of the greenhouse effect) asis shown in Table 2.1 (e.g. Hunten et al., 1983). Clouds on Venus are made mostlyof sulfuric acid and they help distribute the heat around the planet, and because of thisthe temperatures on both the day-side and the night-side are almost equal. As shown inFig. 2.3 the main cloud layer lies approximately in 45 km above the surface and is about15 km thick – above and below this the atmosphere is quite clear and just sparse haze isapparent. The clouds rain sulphuric acid but this does not reach the surface because theintense heat below the clouds evaporates the raindrops about 30 km above Venus.

Figure 2.3: Structural and thermal profile of venusian atmosphere [i3]

The wind is very fast in the upper atmosphere (around 350 km·h−1, almost 60 timesfaster than the rotation of the planet; this is called superrotation) but at the surface is quiteslow (around 5 km·h−1). All around the planet the wind direction is the same as the planet’srotation direction but it is slower closer to the poles, which gives the clouds a typical ”V”pattern. At altitudes greater than 100 km the day-to-night atmospheric flow is observed.


During the Venera and Pioneer Venus missions plasma wave signatures similar to theones on Earth caused by lightning (whistlers) were detected. But the question was if thesesignatures were of local origin (in the neighborhood of the space probes) or they weregenerated in the atmosphere of Venus. The latest observation made by the Cassini spacecraftduring two close flybys in 1998 and 1999 failed to detect any high-frequency radio waves(spherics) also commonly connected to the lightning (Gurnett et al., 2001). But therestill remains a possibility of an other lightning, e.g. from clouds to ionosphere – this producesslow discharges and such low-frequency radio signals that they are difficult to detect.

2.1.3 Surface

The surface of Venus has mainly an unimodal character and over 80% of it lies within ±1 kmfrom planetary mean radius. Unimodality is characteristic for all terrestrial bodies in theSolar System except Earth where the topography is basically bimodal at levels of continents(+200 m) and sea-floor (−3500 m). A comparison of planetary modalities can be seen inFig. 2.4 left – at right is a cumulative age distribution of surfaces.

heig

ht [k

m]

-6

-4

-2

0

2

4

6

Venus

Earth

Moon

Mars

cumulative area higher than h [%]0 20 40 60 80 100

cum

ula

tive a

rea

younger

than

[%]

t

t [Gyr]4 3 2 1 0

0

50

100

high-lands

volcanicplains

Venus

Earth

Moon

Marscontinental

crust

oceanic

cru

st

Figure 2.4: Modality and cumulative age distributions of terrestrial planet surfaces (bothafter Schubert et al., 2001)

Most of Venus’ surface (Fig. 2.5) consists of relatively flat volcanic plains with severalbroad depressions and a few highlands. Everywhere on the surface we can see only a littleerosion because of the absence of water and the slow wind at the surface. The denseatmosphere also prevents small meteors from impacting so only medium and large cratersare observed.

The venusian highlands are 3-5 km above the average surface level – at the equatorAphrodite Terra (size of South America) and Beta Regio (smaller volcanic region), southernAlpha Regio and small Lada Terra. Close to the north pole of the planet is Australia-sizedIshtar Terra which shows several interesting features. Here, next to each other are locatedMaxwell Montes – the highest peak of Venus (10 km) – and several tectonic-like geologicalunits. For these features some authors have proposed a crustal folding mechanism basedon lithospheric stresses (e.g. Sandwell et al., 1997) whereas others favor as their causedynamic-mantle topographic support, which could be a generating mechanism for bothhighlands (including local upwellings (Grimm and Philips, 1991) as well as downwellings


(Bindschadler et al., 1990)) and lowlands. There is no obvious evidence for ongoingor remanent plate tectonics similar to the subduction zones or large rifts located on Earth(e.g. Kaula, 1994). A possible reason for this could be the lack of water on Venuswhich enables the lithosphere to sustain larger stresses and so prevents the creation of platetectonics (section 2.3). Thermal exchange between the planet and space is very efficientthrough plate tectonics and therefore an interior of the planet with a single-plate surfacewould slowly warm up (Nimmo, 2002). However we can find a lot of unusual topographicfeatures on Venus (besides volcanoes – Fig. 2.6 – and craters – Fig. 2.7) like coronae andarachnoids (Fig. 2.8), chasmas (Fig. 2.9) and tesserae (Fig. 2.10) – some of which couldbe areas of local subduction. These together with signs of recent volcanic activity (lavachannels – Fig. 2.11 – and domes – Fig. 2.12) show that Venus could still be an activeplanet. The surface itself (Fig. 2.13) is basically basaltic but more detailed geochemicalresearch is needed in the future.

ISHTAR TERRA

APHRODITE TERRA

LADA TERRA

AlphaRegio

BetaRegio

MaxwellMontes

FortunaTessera

LakshamiPlanum

AtalantaPlanitia

ArtemisChasma

DianaChasma

ThetisRegio

OvdaRegio

AtlaRegio

Ozza Mons

Maat Mons

DaliChasma

TellusRegio Niobe

Planitia

TethusRegio

AinoPlanitia

EistlaRegio

BellRegio

UlfrunRegio

ImdrRegioLavinia

Planitia

HelenPlanitia

DevanaChasma

PhoebeRegio

ThemisRegio

MethisRegio

GuineverePlanitia

Rhea Mons

Theia Mons

Figure 2.5: Topographic map of Venus based on the Magellan’s observations [i3]


Figure 2.6: Magellan radar image of Sapas Mons volcano [i1]. On Venus there are severallarge shield volcanoes (similar to Hawaii or martian Olympus Mons) and many small oneswhich could be sources for the lava covering a large part of the surface. Recently announcedfindings indicate that Venus is still volcanically active, but only in a few locations – mostof the surface has been rather quiet geologically since the resurfacing event

Figure 2.7: Magellan radar image of Wheatley crater [i1]. Craters at Venus are generallycircular and sometimes surrounded by a rim formed mainly by meteorite impact. Howeverwe do not find the abundance of small craters as we do on the Moon or Mars because thesmall meteorites are destroyed during the flight through the dense atmosphere. But for thesame reason we find there groups of craters – the dense atmosphere breaks the meteoritesinto smaller parts which impact around the same place


Figure 2.8: Magellan radar image of a corona and arachnoids [i4]. Coronae are circularsurface features which are usually surrounded by several concentric ridges. We suppose thatcoronae are formed by hot spots in the mantle of Venus or by collapsed domes over largemagma chambers. Arachnoids are smaller but similar to coronae, often clustered and withradial fractures and inter-fractures

Figure 2.9: Magellan radar image of landslide debris in Devana Chasma [i1]. Theseformations and other similar features on the surface are basically deep valleys with oftenserpentine structure and are found mainly near volcanic or proposed-tectonic features. Oftenthey are interpreted as signs of rifting however the originating mechanism is not yet fullyunderstood


Figure 2.10: Magellan radar image of tessera in Eistla Regio [i1]. Tesserae are complexridged terrains situated usually in the plateau highlands of Venus. They are quite uniquebecause we suppose the mechanism of their origin is similar to the tectonic mountain-building processes on Earth, despite that today there are no signs of the plate tectonicson Venus. So this thickening and folding of the crust should originate through some othermechanism

Figure 2.11: Magellan radar image of a channel close to Fortuna Tessera near to IshtarTerra [i1]. The origin of these channels is not clear however we can dismiss that they wereformed by liquid water – the only possible fluid which could have caused them is a low-viscous lava. They can extent up to 1,000’s km (much longer than similar ones on theMoon) and with widths 100’s-1000’s m. These features seem to be some of the most recentat the surface as they show no or little erosion and few superimposed craters


Figure 2.12: Circular domes on the eastern edge of Alpha Regio [i4]. These volcanicfeatures seem to have an origin in eruptions of high viscous lava – they are often locatednear volcanoes or coronae which are both probably manifestations of ascending plumes. Thedomes have diameters up to 10’s of km, steep sides and usually complex fracture patterns

Figure 2.13: Pictures from Venera 9 and 14 (top and bottom respectively) which landedon the surface of Venus [i5]. The first panoramas from Venus were sent to Earth by therussian spacecraft Venera 7 in 1971. We notice the significant differences between the twolanding sites – the first of them represents a tectonic site whereas the second one is typicalof volcanic plains


2.1.4 Crust, Lithosphere and Mantle

There is not much information on venusian interior structure as we lack any seismic datafrom surface landing probes. Because of this we must estimate the mantle structure basedon our knowledge of Earth and from the application of the appropriate scaling law.

The crustal and lithospheric thicknesses are derived usually from observed gravity signalsand topography by means of gravimetric inversion combined with crustal isostasy and flex-ure. Many authors (e.g. Arkani-Hamed, 1996; Simons et al., 1997; Watts, 2001)put the bottom of the crust at the depth of between 15-50 km. However for the bottomof lithosphere the values differ much more – the estimates put the bottom of the litho-sphere between 50-300 km but the elastic lithosphere seems to coincide with the crust (e.g.McKenzie, 1994; McKenzie and Nimmo, 1997). Some others (Simons et al., 1997),on the other hand, doubt our ability to discover the lithospheric thickness from data whichwe have presently – also from this point of view the information from upcoming missions toVenus might be crucial.

Just as the information we have on the subsurface is limited, our knowledge of mantleproperties is very poor. However using knowledge of the Solar System’s initial conditionsand using thermal evolution models Stevenson et al., 1983 derived the depth of mantleto be 2900-3200 km. From a scaling law and knowledge of Earth the phase transitions(endothermic and exothermic) are assumed to be at depths of 440 and 740 km respectively(e.g. Schubert et al., 1997).

2.1.5 Core

As mentioned in Section 2.1.1 Venus does not possess an intrinsic magnetic field. That givesa strong condition on venusian core structure because we assume an internally generatedfield is produced by the thermal convection in the liquid, electrically-conductive part of core.

One possible explanation of this is that the core is completely frozen. In the time of theSolar System’s formation the more volatile elements like sulfur were partially blown out fromthe Sun’s neighborhood and so Venus’ interior contains less iron sulfide than the Earth’s one.This lighter alloy decreases melting temperatures and since the venusian interior conditionsare not sufficient to melt the core, it could stay solid. However the observed venusian meandensity is slightly less than Earth’s (97% of Earth’s density – e.g. de Pater and Lissauer,2001) which contradicts the supposed smaller amount of these lighter materials in the core.

Another theory by Stevenson et al., 1983 states the possibility of a liquid core butwith no convection. The slow rotation of Venus induces a Coriolis force great enough tokeep the convection but the thermal gradient dropped down during the planet evolutionand the magnetic field disappeared approximately 1.5 Ga. And because there is no solidinner-core the energy from phase transition (driving the thermal convection in Earth) ismissing. This theory is supported by venusian Love number estimates which indicate thatthe core is liquid (Konopliv and Yoder, 1996). But as the planet cools it is possiblethat the solidification of the inner-core would happen (Fig. 2.14) and the magnetic field willbe generated again.


Figure 2.14: Possible formation and evolution conditions for cores of Venus and Earth (afterStevenson et al., 1983)

2.2 Evolution of Venus

Venus was formed during the accretion procedure of Solar system circa 4.5 Ga (giga-yearsi.e. billion years ago). Because of its proximity to the Sun a relatively major part of thevolume is constituted by heavy elements (Fe, Si, Al etc.) with additional of volatiles (N, O,H2O etc.) as in the case of Earth. From this comparative view most of our ideas about theevolution of Venus were originated.

In the beginning the planet was undifferentiated but the gravitational force and initialheat (from impacts and radioactive decay) caused the heavier elements to sink to the centralpart and the lighter ones to ascend to the surface. This process formed the core and alsothe initial mantle rheology – both of them then evolved as the primitive structure changed.The interior cooled and was chemically transformed and so the current state is a result ofboth the initial condition and planetary evolution.

2.2.1 Resurfacing Event Constraint

Even if we do not know the exact history of venusian evolution we can make judgementson it based on observed features. One of the most striking ones is the relatively low craterdensity observed by Magellan radar on the surface, as can be seen on comparison diagramsin Fig. 2.15. This means that any record of the craters older than approximately 500-750Ma (Schaber et al., 1992; McKinnon et al., 1997) was erased by some global processand only the younger ones can be observed.


In any model of Venus’ interior there must be a possibility of some unique event whichchanged the surface of the planet at a global level. The most suitable explanations isprobably a resurfacing event which caused a subsidence of the existing surface (e.g. Tur-cotte, 1993) or covered it completely by material from volcanic eruptions material arelatively short period (e.g. Nimiki and Solomon, 1994). A possible mechanism for thefirst of them is a model of catastrophic lithosphere instability which could be a result ofa lithosphere-thickening process connected with a single-plate tectonics. For the second,authors speculate about a periodical convective style, which once per period longer than500 My, causes enormous volcanic activity. Of course the possibility of plate tectonics andvolcanic suspension is an alternative which we can not deny. In all cases the most importantfactor could be the lack of water which had a major influence on the mantle rheology – thedrier subsurface structure could be also the reason for the single-plate tectonics.

2.2.2 Heat Transport Models

It is very important is to fully understand the heat transport process. In the case of Earthabout 70% of the internally generated heat is transferred by subduction of cold lithosphericslabs and by production of a new and hot oceanic crust at the middle-oceanic ridges. Otherminor contributions are by ascending hot-spot, lithospheric delamination and secondaryconvection. However in the case of Venus we observe no plate tectonics and so the heattransfer must be done by some other means. There are generally three possible ways (seereview in Schubert et al., 2001): the uniformitarian model (with constant heat flux andstable internal conditions), the catastrophic model (the heat flux is not in balance with heatgenerated and stored in the planet) and the differentiated-planet model (all heat producingelements are concentrated in the crust).

Figure 2.15: Crater density statistics for three well explored objects of the Solar System –the planets Venus, Earth and Mars. For Venus and Earth we see (except for the protectiveinfluence of atmosphere on small-size meteor penetrations) a certain degree of erosion butfor Mars there are parts of the surface saturated with almost no erosion (after de Paterand Lissauer, 2001)


2.2.3 Convection Models

The very efficient means of cooling the deep planetary interior is the subsolidus mantleconvection. This process controls not only the evolution of Earth (where it is demonstratedby plate motions) but also the thermal and chemical structure of other terrestrial planetsand moons (Schubert, 1979; Schubert et al. 1986) where the manifestation is merelyindirect, through surface volcanic and tectonic features. From our knowledge of Earth andpresumed initial structure of Venus there is a general agreement that this planet is probablystill active. Therefore through modelling of mantle convection we obtain important insightsinto the physics of Venus as well as its evolutionary history.

Several numerical studies of convection in the venusian mantle have been carried out– both for Cartesian (e.g. Steinbach and Yuen, 1992) and for 3D spherical geometry(e.g. Schubert et al., 1997). The results show that the parameters of the model havea significant influence on the convection pattern, degree of layering, state of mixing andlong-time behavior of convection – an example of the importance of an increase in viscosityis given in Fig. 2.16. Aside of this the importance of upper boundary condition is considered– whether the rigid or free upper boundary (stagnant, sluggish or mobile lid) is the case forVenus. If great importance in the mantle behavior is a Rayleigh number, which controls thenature of convection (e.g. Ratcliff et al., 1997) in all the parametrization – for moredetailed explanation see an extensive review on this theme in Schubert et al., 2001.

Figure 2.16: Convection models for Cartesian box 8 × 8 × 1 with temperature-dependentviscosity and fixed Rayleigh number – from left to right and up to down we have viscositycontrasts: 100, 101, 102, 103, 104 and 105 respectively (redrawn from Schubert et al.,2001). From comparison with observed dynamic features on the surface of Venus we candeal with viscosity models a)-d) and reject e)-f) for its small scale character


But another approach to the study the venusian mantle dynamics is also possible –because of the assumption, mentioned several times already, that Venus’ topography andgeoid are manifestations of interior dynamic processes we can include the gravity, topographyand admittance as constraints for quantitative mantle convection. From such global models(e.g. Kiefer et al., 1986) we obtain surprisingly good agreement with observed valuesfor even very simple parametrizations. Also from these studies come our current estimatesof viscosity structure and mantle rheology – see the discussion in Chapter 7. A similartechnique could be used also on the regional scale for those major surface features withstrong gravity signals e.g. venusian equatorial highlands (Kiefer and Hager, 1991) orlarge volcanoes (Kiefer and Hager, 1992) but with the limitation that the obtainedresults may not represent the global characteristics. On the other hand the large percentageof explained observed data allows us to use these forward problems results to consider thevenusian mantle as still being active, with similar structure to Earth, except for the existenceof an asthenosphere and with a stiff lithosphere on top.

2.3 Quest for Water

As has been mentioned several times throughout the text the key question for understandingVenus is how much water it originally had and where it has gone. When we know the answerto this question we can assume more clearly the ongoing geophysical processes within thevenusian interior and also the evolution of its climate conditions. Mainly the convection styleproblem and mantle rheology are tightly connected to the water budget in the minerals andthese could have a significant influence on the resurfacing event.

Our only means of studying these problems are through studying surface mineralogiccomposition or atmospheric element ratios. Unfortunately the first is limited to the datafrom russian Venera missions and has not given us satisfactory answers. As our only otheroption the second possibility strongly depends on exact measurements not only of the ele-ment composition of the atmosphere but also of the escape-time from the atmosphere intospace. This problem is already mentioned in Section 2.1.2 where the possible explanationsfor the current hydrogen and deuterium contents are also noted. However to decide if theD/H ratio is a sign of relatively late outgassing or a residual sign of original water reservoirswe need to have exact measurements of H escape flux (Donahue, 1999). Because thesevary in the order of magnitude (7× 106 - 3× 107 cm−2·s−1) the originating water could mayhave evaporated from the surface anytime from 500 Ma to 4 Ga. To decide whether Venushad an ocean or not we have to improve the sensitivity of our measurements.

2.4 Comparison with Earth

As mentioned above Venus is in many ways similar to Earth. In Table 2.2 some of thephysical parameters and ratios of both planets are shown for comparison. The first threeorbital parameters are directly connected to Venus’ position in the Solar System, the otherparameters probably to the evolution.


planetary parameters Venus Earth ratio

mean distance from Sun 108 mil. km 150 mil. km 0.72

orbital period 225 day 365 days 0.66

orbital velocity 35 km·s−1 30 km·s−1 1.18

rotational axis inclination 177.4 23.5 –

rotational period 243 days 1 day 243

mass of planet 4.87×1024 kg 5.98×1024 kg 0.95

fraction of atmosphere to total mass ∼ 10−4 ∼ 10−6 100

mean density 5.24 g·cm−3 5.52 g·cm−3 0.95

mean gravitational acceleration 8.89 m·s−2 9.81 m·s−2 0.91

escape velocity 10.36 m·s−1 11.2 m·s−1 0.93

albedo (reflexivity) 0.65 0.37 1.76

temperature interval +446 to +482 C −88 to +58 C –

number of moons 0 1 –

Table 2.2: Comparison of Earth and Venus (after de Pater and Lissauer, 2001)

According to the latest celestial mechanics studies the orbits of terrestrial planets arevery stable over astronomical short period. But over longer periods they can become chaoticand thus unpredictable because of weak but random-like gravitational impulses from otherSolar System objects. The Ljapunov exponent of orbital decay in the case of Venus isapproximately 50 My−1 (de Pater and Lissauer, 2001) so after this time the orbitchanges distinguishably. But from similar studies done on Earth it is known that thegroup of possible orbits is quite narrow and so the changes in the insolence (important fortemperature trend) are not significant.

Chapter 3

Data Sets

In the previous chapter we saw the importance of exact measurements of topography andgravity for the understanding of internal structure and processes in the mantle when we missany seismic data. Unfortunately such data coverage is available only for a few terrestrialbodies (Venus, Earth, Moon, Mars) because a probe on stable orbit for a relatively longperiod is needed. For the other planets and moons we have just flybys (Mariner 10 aroundMercury, Galileo around Jovian moons) which are not sufficient for the construction ofglobal models.

The planetary topography is measured usually by laser altimetry whereas the gravitymust be derived from the Doppler shift of radio signals received on Earth. These raw datacould be easily converted to the spherical harmonic (SH) model – the spectral approach hasseveral advantages so even the topography is used to transform to the SH model. But be-cause of coefficient normalization by planetary parameters (Kaula, 1966; see Appendix A)proper knowledge of these parameters is needed. In Table 3.1 their values are listed – but asany data from the space missions their values could be re-set again in the future. However,throughout this work we will be dealing SH coefficients with no normalization.

GM[km3·s−2] radius[km] g0[m·s−2]

Venus 324,858.592 6,051 8.89

Earth 398,600.442 6,371 9.81

Moon 4,902.801 1,738 1.62

Mars 42,828.372 3,396 3.72

Table 3.1: Planetary parameters for SH coefficients normalization [i4]

One of the reasons for the use of SH models is to see the power of topography or thegeoid on different scales – from this we can assume certain implications about the internalstructure of planet. However these spectral methods have of course global characteristicsand therefore they are unsuitable for the investigation of individual features on a local scale.

27

CHAPTER 3. DATA SETS 28

3.1 Spherical Harmonic Models for Venus

Both venusian geoid and topography models are known from orbital measurements madeby the spacecrafts Pioneer 12 Venus Orbiter (1978–1993) and Magellan (1990–1994). Thelatter data sets from Magellan are much more accurate and so we have used them in ourinvestigation. However because of a strong coupling between the present topography modeland new gravity measurements and vice versa there is still a chance for future improvements.The SH coefficients were downloaded from Geophysical Node of Planetary Data System [i4].

3.1.1 Geoid model MGNP180U

The model of Venus’ geoid was derived from Doppler tracking of the Magellan orbiter inform of line-of-sight (LOS) acceleration measurements. All aerodynamic processes weremodelled so the gravity acceleration could be read from the Doppler shift of two-way radiocommunication and converted into a geoid model (Konopliv et al., 1999). The first threeorbital cycles (1 cycle ∼ 243 days) of the Magellan mission were used just for preliminarymodels. The collection of data began with cycle 4 and continued until the loss of thespacecraft late in 1994. Using an equation A.30 we can convert measured quantities to agravitational potential – this then gives us then the geoid as is shown in Fig. 3.2.

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Figure 3.1: Power spectrum of the venusian geoid displayed an a linear and logarithmicscale compared with Earth geoid power spectrum (using equation A.12)

3.1.2 Topography model GTDR.3

During the Magellan mission orbital cycles 4, 5 and 6 4,284,578 measurements by its laseraltimetry device were obtained, and so the entire surface was covered in great detail severaltimes. The radial position of surface elements was determined based on gravity modelsobtained from previous maneuvering of the Magellan itself (see section above). The map inFig. 3.3 shows 0.25 resolution of topography in equidistant projection – the correspondingSH model (Rappaport et al., 1999) was completed up to the degree and order 360.


Figure 3.2: Model MGNP180U of the venusian geoid. The dashed line marks the zero-leveltopography contours

Figure 3.3: Model GTDR3.2 of venusian topography


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Figure 3.4: Power spectrum of the venusian topography displayed in linear and logarithmicscale compared with Earth topography power spectrum

The mean planetary radius was defined with new precision as 6,051,881 m and theplanetary flattering of Venus as 9.75× 10−5. New calculations were also done for center-of-figure offset from center-of-mass and it was found lying beneath the north-west corner ofThetis Regio in Afrodite Terra with the radius of offset rf = 186.5 m. This position is causedmainly by equatorial highlands which, given their non-uniform distribution mainly aroundthe equator, mark an accumulation of the surface mass. However when compared to Earth’stopography Venus’ is generally much flatter, which is obvious also in the power spectracomparison diagram in Fig. 3.4. This could again corresponds to the overall character of thevenusian surface (see Section 2.1.3) and our assumption that a large part of it represents the

-2000

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Figure 3.5: Displacement of digital and spherical harmonic models of venusian equatorialtopography with respect to the reference radius (after Rappaport et al., 1999)


dynamic topography which has to have much smaller amplitudes (e.g. Schubert et al.,2001) than a plate-tectonics one, with its existence of continents, sea-floor with subductionzones and plate-collisions areas inducing high mountain-chains of tectonic origin.

As mentioned above this latest spherical harmonic model of venusian topography isevaluated up to degree and order 360. It shows a generally higher degree of correlation withthe geoid – in comparison to previous models – over the entire spectral interval and alsofor short wavelengths in spatial domain (Konopliv et al., 1999) and so enables us toinvestigate surface features of a smaller scale (∼ 10’s km) objects like coronae or volcanicdomes. In Fig. 3.5 the misfit between the spherical harmonic model and the real digitalaltimetry on the equator is displayed (please note 2500× vertical exaggeration). Howevera priori constraint of using the Kaula’s rule of thumb for higher degrees of geoid model(Konopliv et al., 1999) is a reason for the decreasing of correlation coefficient (Fig.3.6 left) and for change of the admittance function character (Fig. 3.6 right) for degreesj > 100. Therefore we will use in our calculations both models truncated at j = 90 whichguarantees the certain independence of data sets.

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itta

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Figure 3.6: The degree correlation between Venus’ geoid and topography (A.13 – the dashedline marks a 95% level of confidence) and degree admittance (A.15)

The key question however is how the topographic load is supported. In following twochapters we will apply different approaches to answer this question and by used methodsthen predict the venusian geoid. By comparison of the observed data and the predictedvalues we can find out properties of the venusian subsurface.

3.2 Spherical Harmonic Models for Other Planets

As we are enhancing our knowledge of the Solar System with automatic space probes map-ping individual planets and moons we are still strongly pushed to look for the parallelsbetween these bodies and our own planet Earth. Unfortunately the progress is complicatedand several missions to a planet are needed to obtain reliable data with sufficient resolution.Some basic gravity and topographic data for Earth, Moon and Mars are shown to under-


lay wider planetological considerations which could help us not only in planetary researchbut also back in the Earth sciences. The data used (SH coefficients or grided values) wereobtained from [i4] for Moon and Mars whereas those for the Earth from [i6] and [i7].

In Fig. 3.7 are the observed geoid and topography of Earth together with the powerspectra and their correlation. The purpose of showing these diagrams is to offer a chanceto compare it with other terrestrial planets – e.g. the correlation diagrams show importantglobal-scale difference in the nature of planets. This is especially the case when compared toVenus, where the correlation coefficient for j > 2 is above 95% level of confidence. Anotherinteresting feature is the bipolar structure of the geoid which is, in the case of Earth, inter-preted as a manifestation of D” heterogeneity reservoirs connected with mantle upwelling(e.g. Forte and Mitrovica, 2001) but is similar to a martian geoid dominated by theTharsis region and Utopia Planitia (which is by some – e.g. Harder and Christensen,1996 – presumed to be a remanent sign of single-cell convection in the mantle).

Despite the fact that the Moon is our closest space neighbor we had to wait until 1990’sto obtain more detailed information – the Clementine (1994) and then Lunar Prospector(1998–1999) missions provided us the global gravity and topography data. Recent gravitymodels have been completed up to the degree and order 165 (Konopliv et al., 2001) butwith the handicap of no direct information from the far-side which limits the informationon wavelengths l > 110 and needs a strong a priori constraint. From laser altimetry severaltopographic models were completed – here we use the model up to degree and order 90(Smith et al., 1997) however truncated at j = 70 because of the coverage gaps in thepolar regions. In the close future a major advance in lunar research is expected so theseboth problems should soon be solved and complete SH models up to a higher degree will beavailable. Analyzing these fields (Fig. 3.8) we can immediately observe the most strikingfeatures in the lunar geoid: so-called mascons (mass concentrations). Most of these stronggravity peaks are located in large Mares or craters where the topographic minima are andmost probably originate in some subsurface mass anomalies.

In the case of Mars the main contribution to our knowledge of the planet came frommission Mars Global Surveyor (1997-until now). This spacecraft’s altimetry device produceda global topography model (Smith et al., 1999) with better precision than we have formany locations on Earth. However the current model of martian geoid GMM-2B (Lemoineet al., 2001) is only up to the degree and order 90 and in the short wavelengths a strong”contamination effect” is observed – see Fig. 3.9. Because of this basic problem the scientificcommunity is impatiently awaiting the results from new missions, namely from the europeanMars Express which has been in a polar orbit around Mars since 2003. The new results willthen be used to better determine the crustal properties (of which the global distributionis already know from gravimetric inversions done recently by Zuber et al., 2000) andparameters of the lithosphere and mantle which are connected to planetary evolution.

In the future the automatic spacecrafts will explore for us the planet Mercury, the largestbodies of the asteroid belt and the Jovian moons. From these data we will hopefully uncovermore on the Solar System’s formation, terrestrial planet evolution and possibly also on icy-moon structure. The latter is crucial both for planetary sciences and for exobiology as weassume there are water oceans under the icy-crusts (Spohn and Schubert, 2003).


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Figure 3.7: Earth’s geoid (top) and topography (middle) together with their power spectraand the correlation between observed geoid and topography


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Figure 3.8: Lunar geoid (top) and topography (middle) together with their power spectra(compared to Earth – blue lines) and the correlation between observed geoid and topography


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Figure 3.9: Martian geoid (top) and topography (middle) together with their power spectra(compared to Earth – blue lines) and the correlation between observed geoid and topography

Chapter 4

Isostasy and Crustal Structure

In Chapter 3 we saw in the correlation and admittance diagrams, that the key point inunderstanding the mechanisms producing the topography is an interpretation of the ac-companying geoid. With Venus this is well correlated to the topography, hence the easiestexplanation based on steady-state processes is acceptable. If these predict the observed datawith high accuracy the current status of Venus may be explained by already ceased dynamicmantle processes and the heat flux removed from the planet through a thermal conduction.

4.1 Uncompensated Topography

Any model of the venusian geoid based on steady-state processes should contain the con-tribution of an uncompensated topography – this case is described by the equation A.26.However, comparing the observed and predicted power spectra (Fig. 4.1) we see that such amodel predicts a geoid with a power spectrum of two orders of magnitude higher than theobserved data.

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Figure 4.1: Power spectrum of geoid induced by the uncompensated topography comparedwith the power spectrum of the observed geoid

37

CHAPTER 4. ISOSTASY AND CRUSTAL STRUCTURE 38

Figure 4.2: Geoid induced by venusian uncompensated topography

From analysis of the predicted power spectrum we can reject the hypothesis of uncom-pensated topography. Nevertheless it is worthy to also examine the spatial expansion ofthe obtained SH model. As seen in Fig. 4.2 we obtained a geoid with amplitudes one orderof magnitude higher than the observed data. The major features of this model are alsonot well correlated with the observed ones as could be seen in comparison with Fig. 3.2 atpage 29 (e.g. a high signal at Aphrodite Terra but instead of a major peak in Atla Regiothis is quite low and higher signals are located in Ovda Regio – Maxwell Montes in IstarTerra became the highest peak on the planet). Because of these reasons we have to look forsome other geoid generating mechanism.

4.2 Isostatic Compensation of Topography

To diminish the topographical contribution to the geoid we can employ the principle ofisostatic compensation to our model. For our purpose we adopt a simple scheme of ahomogeneous crust with density ρ = 2,900 kg·m−3 (e.g. Rappaport et al., 1999). Tosatisfy a principle of isostasy we prescribe on the bottom of the crust a topography which isan inverse of the surface and having amplitudes multiplied by the ratio

ρsurf

∆ρmoho. We cannot

determine the density contrast at the bottom of the crust ρmoho, but when we are onlydealing with the geoid contribution from this boundary, the amplitude of it cancels thisterm and we obtain the equation A.28 which is independent from it.

First we look for a single apparent depth of compensation (ADC) in the whole spectralinterval j=2-90 (which is not severely biased by a priori constraint). We vary the parameterof the compensation depth in the interval 1-500 km and for every depth we calculate the


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Figure 4.3: Power spectrum of the geoid induced by topography compensated in 165 kmcompared with the observed geoid power spectrum (left) and a misfit function for compen-sation in the whole spectral interval j=2-90 (right)

Figure 4.4: Geoid induced by topography compensated in 165 km. The predicted geoidshows similar defects to the uncompensated one, especially overcompensation at Istar Terra(twice the signal of the observed data) and Ovda Regio


least-squares misfit (based on A.29) between our model’s predictions and the observed data.In the misfit diagram (Fig. 4.3) we see that the minimal misfit is obtained for a global ADCof 165 km. Such a crustal thickness model, however, produces a power spectrum (Fig. 4.3)and a geoid (Fig. 4.4) which are not in close agreement with observed data.

The second hypothesis which we will test is the assumption that only small scale venusiantopography is isostatically compensated. Hence we apply the previously used method forfinding a global ADC separately for each degree in the SH model (Arkani-Hamed, 1996).In the resulting diagram (Fig. 4.5 left) we can see the optimal depths of compensation forsurface features at all wavelengths. At degrees j=2-40 the ADC is decreasing more or lessmonotonously from 200 to 30 km. But for degrees j > 40 the global ADC stays almostconstant – from a misfit minimization (Fig. 4.6 right) we obtain a value of 35 km. Thereforewe can suppose that the features smaller than approximately 500 km are well compensatedby the crustal isostasy. Another approach to this problem (Simons et al., 1997) is basedon fitting the admittance curve (Fig. 4.5 right) – calculated by using the equation A.15– by the theoretical curve for the compensated topography. As shown, for degrees 40-90a predicted admittance for compensation depth 35 km fits well with the observed values.

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Figure 4.5: Degree ADC and degree admittance between the geoid and topography withsynthetic admittance curves for 35 (red line) and 165 km (blue line) thick crust.

When we are studying the resulting power spectrum (Fig. 4.6 left) and geoid (Fig. 4.7)we find no direct contradictions with observed data and therefore this model of isostatictopographic support is acceptable. However, the amplitudes of both compared quantitiesare smaller than observed ones. From that we can assume that the geoid generating mech-anism is not only of isostatic nature but rather some complex mechanism with a possibleisostatic contribution from shallow compensation. This judgement then implies the possibleimportance of some active processes in Venus interior (presumably in the mantle) which willbe studied in the next chapter.


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Figure 4.6: Power spectrum of the geoid induced by topography compensated in 35 km com-pared with the observed geoid power spectrum (left) and a misfit function for compensationin the spectral interval j=40-90 (right)

Figure 4.7: Geoid induced by topography compensated in 35 km. We can see that biggestcontributions are in the tectonic or volcanic areas such as Ovda Regio, Ishtar Terra or MaatMons but these should be taken only as minor components of the observed field


4.3 Degree ADC Method for Other Planets

Because the used method of degree ADC (as well as the corresponding method of degreeadmittance) is simple and in the case of Venus gives us a reasonable results we would liketo see if it also has a general validity for other planets. Fortunately as can be seen on thenext page this method more or less also works well for the Earth and Moon and thereforedoes not cast doubt on our results obtained for Venus.

For Earth the situation is similar to Venus. When we study the geoid induced by anuncompensated topography (e.g. Rozek, 2000) we obtain a field with amplitudes oneorder of magnitude higher than the observed one. From seismic experiments we know thatMohorovicic discontinuity is situated approximately 30 km beneath the continental surfaceand less than 10 km beneath the sea-floor. Looking at the degree ADC diagram (Fig. 4.8left) we see that our results (at degrees j > 50 the global ADC between 15 and 25 km)are somehow averaging these values and therefore are in agreement with the observations.The admittance diagram (Fig. 4.8 right) confirms this – it also shows interesting negativeadmittance values for long wavelengths which are probably caused by the anticorrelatinggeoid and topography.

The size of the Moon is half of the size of Earth or Venus but the anomalies of the geoidare of one order of magnitude higher (Fig. 3.8). The main contribution to its gravitationalpotential is from subsurface mass concentrations and impact basins – moreover from theApollo seismic experiments and mare observations we assume a near-side/far-side dichotomywhere the crust of the far-side could be twice as thick as of the near-side. The uncompensatedtopography model predicts similar values to the observed ones suggesting that the maincontribution to geoid is presumably not of dynamic origin. When we carry out the degreeADC method (Fig. 4.9 left) we find for degrees j > 20 a compensation depth of around 20 kmwhich is in agreement with the latest processing of the Apollo Passive Seismic Experimentwhich gives a compensation depth of around 30 km. However the oscillations of ADCdiagram as well as some anomaly in the admittance curve at degree j=10 (Fig. 4.9 right)signalize a strange behavior of lunar gravity and topography.

Finally we applied the methods of degree ADC and admittance to the planet Mars. Nev-ertheless because a strong alias effect in the geopotential model GMM-2B (and maybe alsobecause of the major surface features – the giant Tharsis uplift and nort/south dichotomy)we are not able to determine the global ADC value. However from local values the crustalthickness ∼ 50 km has already been derived (Zuber et al., 2000).


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1 1010 100

degree

Figure 4.8: Degree ADC and admittance function in the case of Earth

0

50

100

150

AD

C [km

]

0 20 40 60 80 100

degree

-20

0

20

40

60

80

100adm

itta

nce [m

/km

]

1 1010 100

degree

Figure 4.9: Degree ADC and admittance function in the case of Moon

0

100

200

300

400

500

AD

C [km

]

0 20 40 60 80 100

degree

0

50

100

150

200

250

adm

itta

nce [m

/km

]

1 1010 100

degree

Figure 4.10: Degree ADC and admittance function in the case of Mars

Chapter 5

Dynamic Model of Geoid

In our effort to determine the mechanism providing the support of observed topography atthe long-wavelength part of spectrum, we focus on the dynamic processes in the mantle, asthey are supposedly still active (e.g. Schubert et al., 2001) and because their numericalmodelling could give us valuable results concerning the inner structure of Venus (Kieferand Hager, 1992; Kiefer and Peterson, 2003).

In mantle convection the common principle driving the whole process are the buoyancyforces induced by thermal (i.e. density) inhomogeneities. The relatively hot and cold partsof mantle induce the flow which is controlled by both density structure and mantle rheology.

+

- -

Figure 5.1: Upwellings and downwellings in the mantle induced by negative (red) and posi-tive (blue) density anomalies. At the boundaries the buoyant forces induce non-zero stressdeforming both the surface and CMB

45

CHAPTER 5. DYNAMIC MODEL OF GEOID 46

The stress associated with this motion of mass acts on any boundary – surface, CMB –of the mantle and deforms them (schematically in Fig. 5.1). The dynamic topographypredicted at the surface could be successfully compared in the case of Venus to the observedone. Moreover the uneven mass distribution beneath the surface together with the dynamictopographies at the surface and CMB induces the variations in the geopotential – whichcould also be compared to the long-wavelength geoid observed. By looking for a model withthe highest agreement with observed data we can learn about the mantle properties.

Our work is based on the internal loading theory which has already been successfully usedfor forward modelling of venusian interior (Kiefer et al., 1986; Kiefer and Peterson,2003). However our approach combines this useful tool with an inverse problem formulationand therefore our results should have a global validity and higher robustness than resultsfrom forward modelling. The extensive description of our approach is given in Appendix C.The major step done by our work was considering only 2D density variations constant withdepth (i.e. ρ = ρ(ϑ, ϕ)) to avoid ambiguity in the solution. Because we have no informationabout the density variations within the venusian mantle, we could obtain the same result(dynamic geoid or topography) for any given viscous profile by using a number of differentρjm models. Therefore to obtain a unique solution we fix the density variations at a valuecalculated to best fit both observed fields. Among such models we then look for those withminimal L2 norm misfit when compared with observed data.

We have adopted two approaches to solve this nonlinear problem – the Monte Carlomethod (random search in the model space) and the simulated annealing algorithm (randomsearch with successive focusing). We consider the mantle with layered viscosity profile (n−1layers with viscosity varying 100-105 and an uppermost layer with fixed viscosity 105). InFig. 5.2 we can see the results for n = 3, 4 (the model for n = 2 was omitted because ofgeneral inconsistency with successive models) obtained by the Monte Carlo method (the 10best from 1,000,000 and 2,000,000 runs respectively) superimposed with the best solutionsreached through the simulated annealing algorithm (red lines).

1e+01

1e+00

1e+02

1e+03

1e+04

1e+05

rel. v

iscosity

0 1000 2000 3000

depth [km]

rel. v

iscosity

0 1000 2000 3000

depth [km]

1e+01

1e+00

1e+02

1e+03

1e+04

1e+05

Figure 5.2: The best viscosity profiles for 3 and 4 layer models of venusian mantle. TheMonte Carlo results (black lines) mark the class of possible solutions whereas for comparisonbetween models we use single results obtained by the simulated annealing (red lines)


-5

-4

-3

-2

-1

0

1

2

3

4

5

log

/h

hup

pe

rlith

o

30 45090 150 210 270 330 390 510

thickness of lithosphere [km]

Figure 5.3: The minimum misfit for various viscosity contrasts (105-10−5) between the uppermantle and lithosphere. The minimal values are blue and the maximal ones are red – the2 layer (60 km) step was used

-5

-4

-3

-2

-1

0

1

2

3

4

5

log

/h

hlo

we

ru

ppe

r

540 780 1020 1260 1500 1740 1980 2220 2460 2700 2940

depth of upper/lower mantle interface [km]

Figure 5.4: The minimum misfit for various viscosity contrasts (105-10−5) between the lowerand upper part of the mantle. The minimal values are blue whereas the maximal ones arered – the 4 layer (120 km) step was used


The number of parameters of our random search is 2n− 1 plus the density distributiongiven by a set of SH coefficients ρjm. However this model vector is not a free parameteras we decided to compute its value analytically to minimize the misfit between observedand predicted values (C.28). To analyze the resolving power of our inverse problem for-mulation we choose the model n = 3. In Fig. 5.3 we can study the resolution of viscositychanges between lithosphere and upper mantle and their positions. Whereas the depth ofthe lithosphere seems to be located relatively well between 100-200 km, the viscosity dropconnected to this remains in a wide interval. The following viscosity change (Fig. 5.4)seems on the other hand to be well detected (increase by 1-2 orders of magnitude) but canbe placed at a wide interval of depths. Therefore we can assume that such an increase isnot a manifestation of a low viscosity zone end but rather of a gradual viscosity rise.

In comparison to the isoviscose mantle model we can reduce, by using the various viscos-ity models, the misfit between the predicted and observed data to about 2/3. However, wecan also consider another parameter of the agreement – the percentage of predicted data p(C.30). For our method and various viscosity profiles we can well predict about 90− 95% ofboth the geoid and topography. However, between individual random-search solutions thedifference in p is insignificant (does not directly depend on n) which could be a consequenceof strong nonlinearity of our inverse problem.

The following statements are made for the solution obtained for n = 4 by the simu-lated annealing algorithm although the difference to other results is rather minor. Thecomparison of power spectra and the correlation between observed and predicted quantities(Fig. 5.5) could be a valuable tool in the evaluation of our findings. The correlation be-tween predicted and observed data is overall very good which supports the dynamic supporthypothesis. The power spectra of geoids and topographies differ slightly nevertheless thepredicted/observed geoid difference is noticeable around j = 10, where we can assume someother minor contribution to the observed geoid.

The degree admittance function was not a constrained quantity but when we comparea prediction with the observed values (Fig. 5.6) we can also see a good fit to this parame-ter. The predicted geoid and topography (Fig. 5.7 left) seems to have the major featurescorresponding well to the observed ones (same figure, at right). However, for precise spatialstudies we have mathematical tools not just based on intuitive comparison – for detailssee Chapter 6. Despite the basic simplification in obtaining the density lateral variationsinducing the mantle flow (Fig. 5.8 left) the uppwelling and downwelling structures do oftencoincide with the assumed mantle structure. This is not only the case of volcanic upliftsites (Atla or Beta Regiones) and major lowlands (Atalanta or Lavinia Planitias) inducedpresumably by ascending and descending mantle flows but maybe also of some highlandslike Ovda or Thetis Regiones (Kiefer and Hager, 1992).


1e+07

1e+06

1e+05

1e+03

1e+04

pow

er

[m]

2

1e+04

pow

er

[m]

2

1e+03

1e+02

1e+01

1e+00

1e-01

1 10 100

degree

1 10 100

degree

1.0

0.5

0.0

-0.5

corr

ela

tion c

oef.

95%

200 40

degree

1.0

0.5

0.0

-0.5

corr

ela

tion c

oef.

95%

200 40

degree

Figure 5.5: The power spectra of predicted dynamic topography (left) and geoid (right)together with observed ones and their degree correlation with observed data. Dashed linesagain denote the 95% level of confidence

0

10

20

30

40

adm

itta

nce [m

/km

]

1 1010 100

degree

50

Figure 5.6: Observed degree admittance (black curve) and predicted dynamic one (redcurve). Good agreement is observed approximately up to degree j = 25


Figure 5.7: Comparison of predicted (left) and observed (right) geoid and topography forwavelengths j=2-40. The biggest difference could be seen in the regions containing tectonicfeatures like surroundings of Ishtar Terra or Aphrodite Terra highlends

101 100

degree

1e+02

1e+01

1e+00

1e-01

pow

er

[kg

m]

2-6

Figure 5.8: Inducing density (j=2-40) for the n=4 result obtained by the simulated annealing(left) and its power spectrum (right) – the power spectrum for j > 40 is also plotted herefor character study (right of the dashed line)


For n ≥ 4 the resulting viscosity profiles contain some instabilities compared to pre-viously obtained trends (Fig. 5.2). The typical one with an inserted stiff layer (Fig. 5.9left) seems to be generated by the nonlinear character of the inverse problem – it appearsincreasingly with higher n and does not have a steady shape but rather is positioning thehigh-viscous layer randomly. However the second characteristic one with a viscosity drop ofsomewhere between 80-100% in the depth of the mantle (Fig. 5.9 right) corresponds to theprevious trends of uniform increase through the mantle. The presence of such low viscositytransitional zone above the core has not yet been reported which can be due a higher sen-sitivity of our global inverse model in comparison to the forward models – however becauseof the low resolution in this deep part of the mantle we cannot accept it unambiguously.But comparing our finding to the fact that Venus presumably still has a melted core whichis just slowly cooling down (Stevenson et al., 1983) we may explain it naturally in theway of a transitional mantle zone above the core.

rel. v

iscosity

0 1000 2000 3000

depth [km]

rel. v

iscosity

0 1000 2000 3000

depth [km]

1e+01

1e+00

1e+02

1e+03

1e+04

1e+05

1e+01

1e+00

1e+02

1e+03

1e+04

1e+05

Figure 5.9: Two kinds of viscose instabilities contained in the set of results for n=4 layermodel obtained by the Monte Carlo random search method

Chapter 6

Localization Analysis

When we need to study some well localized features in the gravity field (e.g volcanic riseor tectonic rift) or to detect some hidden ones (e.g. crustal thickening contributions),appropriate tools to use are the localization methods. In comparison with the spectralmethods – which are for global characteristics – this approach employs the function withcompact support and thus is not affected by distant-zone contributions. There are severaldifferent ways how of realizing it mathematically (see Appendix B) but for our purposes wechose the method recently presented by Kido et al., 2003 (equations B.16-B.21).

6.1 Localization of Data

Using this approach we first of all examine individually the geoid and topography of Venus(applying equation B.18). In Fig. 6.5 left we can see the localization of the geoid for long andintermediate wavelengths – at right there is the corresponding localization of topography.This can give us the first insight into the mass distribution on Venus as well as on the geoid’smajor features. This is the first step in data localization analysis which is needed for otherstudies – all diagrams show a zero-level topography contour.

This method could be used also for the empirical comparison of two fields because thelocalization reveals the real field’s anomalies unaffected by other (global) signals. Such acomparison can also be carried out in the case of localized geoid and topography (Fig. 6.1)but in the next section we introduce a better method for such a task. Instead, we use itto validate our hypothesis of the isostatic support of topography for j > 40. In followingfigures we display a localized geoid for the wavelength j = 40 in the three diverse regions ofIshtar Terra, Aphrodite Terra and the surroundings of Phoebe Regio. As these regions haveanother nature they are obviously also in another isostatic case. Whereas the first of them(Fig. 6.2) seems to be no longer tectonically active and in isostatic equilibrium, for the othertwo this is most probably not the case. The region of Devana Chasma (Fig. 6.3) seems tobe compensated only partially (some authors are arguing a hypothesis that this is a stillan active rift – e.g. Kiefer and Peterson, 2003) whereas for the region west of BetaRegio (Fig. 6.4), where the isostasy does not work well, we have a strong presumption fromcoronae clustering (Squyres et al., 1993) that this could be a site of dynamic activity.

53

CHAPTER 6. LOCALIZATION ANALYSIS 54

Figure 6.1: Localized geoid (left) and topography (right) for wavelengths j = 8, 16, 32(respectively from top to bottom) in cylindrical projection. For shorter wavelengths theresults cannot be displayed as a global view hence the regional studies are preferred. Onthe geoid localization diagrams we can clearly observe the volcanic rises at Atla and BetaRegiones as well as the contribution of the coronae in Eistla Regio. The localization oftopography is revealing the major contributions of the venusian highlands together withIshtar Terra and major volcanoes. At long wavelengths it apparently corresponds to thelocalized geoid which is proven by localized correlation studies in the next section


Figure 6.2: Localization of observed and predicted geoid and corresponding topographicmap (respectively from top to bottom) for wavelength j = 40. The region of interest is inthis case Ishtar Terra with the Maxwell Mountains and adjacent Lakhsmmi Planum (thered lines serve as an orientation guide – they are always in the same position)


Figure 6.3: Localization of observed and predicted geoid and corresponding topographicmap (respectively from top to bottom) for wavelength j = 40. The region of interest isin this case Phoebe Regio and close Devana Chasma (the red lines serve as an orientationguide – they are always in the same position)


Figure 6.4: Localization of observed and predicted geoid (left and right) and correspondingtopographic map (bottom) for wavelength j = 40. The region of interest is in this casebetween Beta and Atla Regiones and Phoebe Regio (the red lines serve as an orientationguide – they are always in the same position)


6.2 Localization of Correlation and Admittance

When we want to compute the degree of agreement between two fields or the transferfunction (admittance) we can employ the common spectral approaches (A.13 and A.15respectively). However this quantity could be quite laterally variable and we would like toseparate the sites with higher values from those with lower one. For these purposes we canemploy the localized correlation (B.19) and admittance (B.20) which come out from thelocalization method mentioned in the previous section.

Localized correlation between geoid and topography (Fig. 6.5 left) clearly proves anoverall good correlation of the two fields on Venus, which is already known from spectralanalysis (Fig. 3.6 left). Moreover we can observe certain regions of the surface which havesome lower correlation for intermediate wavelengths – these specific sites mostly coincidewith some complicated topographic structures. Therefore we can conclude that generallyeven the long-wavelength topography on Venus is a manifestation of internal dynamic pro-cesses. Studying the localized transfer function between geoid and topography (Fig. 6.5right) we can distinguish at intermediate wavelengths several prominent surface featureslike highlands or volcanic raises whereas at long ones we can observe only the general lowervalues of admittance in the sites of plateaus and tesserae.

We can successfully link this tool to our dynamic modelling problem when we use it forthe spatial comparison of the observed and predicted geoid and topography (Fig. 6.6). Forthis purpose we choose only wavelengths j = 16, 32 because of the restriction of dynamicmodel in spectrum and low resolving power at wavelength l = 8. From obtained resultswe can see the general good agreement between observed data and our predictions in theexamined spectral interval. The sites of lower correlation correspond mainly to tectonichighlands where some more complex (not strictly dynamic) mechanisms of origin should beemployed. This could be a significant disadvantage of our dynamic model, however, withthe chosen simplifications (see section C.3 for discussion) we cannot overcome the basicmismatch between the observed geoid and topography (compare to Fig. 6.5 left).

Moreover if one is interested in the properties of a specific site the method of localizedadmittance restricted to this region could be used (for more details see again Simons et al.,1997 or Kido et al., 2003). This approach could be helpful in verification of the laterallyvarying properties as it shows a cross-cut through the entire selected spectral interval forthe surveyed quantity at the site.

6.3 Emax-kmax Method Applied on Venus’ Geoid

When we wish to examine anomalies with maximum energy (i.e. with maximum localizedsignal) at different scales we can employ the localization for each individual wavelength andstudy them separately or we can use a compositional method. In this thesis we applied to theplanetary geoids the Emax-kmax method (equation B.23) which analyzes the contributionsfrom localization at all wavelengths over a given spectral interval and displays the highestvalue at each grid point.


Figure 6.5: Localized correlation (left) and admittance (right) between venusian geoid andtopography for wavelengths j = 8, 16, 32. The admittance function gives reasonable resultsonly in sites of correlation |C| > 0.5 because otherwise it is not clear what the sense of thetransfer function between these fields is. Despite the fact we are using another method acomparable results can be found in Simons et al., 1997


Figure 6.6: Localized correlation between the observed and predicted geoid (top) and theobserve and predicted topography (bottom) for dynamic modelling at wavelengths j=8,16(from left to right). Please note the change in color scheme from Fig. 6.5

In Fig. 6.7 and Fig. 6.8 we can see the Emax-kmax maps of the venusian geoid forlong and short wavelengths respectively – for intermediate ones the result do not differ toomuch from these. However because of the already mentioned good correlation between thegeoid and topography even at long wavelengths the signature revealed by the Emax mapis mainly connected to major topographic features. Surprisingly at the short wavelengthsthe most prominent signal is induced by volcanic rises and highlands and not by deep andlong trenches (e.g. Devana Chasma or Artemis Chasma). The second part of analysis – thekmax map – is strongly influenced by the fact that in the regions of signal sign change eventhe weak short-wavelength is getting the strongest signal. However we can observe severalcomplicated structures in Ovda and Thetis Regiones but the most prominent is locatedbetween Atla and Beta Regiones where the largest cluster of coronae is located (Squyreset al., 1993).

Some authors proposed the possibility that remanent signs of plate tectonics could beinferred by these studies or some subsurface features could be revealed but none of themwere found by our research. If any such hidden structures still exist on Venus, some otherway to detect them must be used, basically after obtaining new data. The most efficientway would be a multiple seismic probing of the venusian interior but other methods are alsoavailable today e.g. sonar probing.


6.4 Emax-kmax Analysis for Other Planets

As we have found the Emax-kmax method useful in detailed survey of the venusian geoid wecan apply it also on the gravity fields of other terrestrial planets.

In the case of Earth a similar analysis done by means of wavelet transform was carriedout by Vecsey et al., 2003. Therefore our findings just confirm their results for regionsnot far from the equator. However our technique enable us also to examine the polarand subpolar regions. In Fig. 6.9 we can see the analysis for long wavelengths which isdominated by Greenland-Iceland, New Guinea and South America positive anomalies andby Asian, Antarctic and North American depressions. The Hawaiian and African hotspotcontributions also appear. On the other hand Fig. 6.10 shows many features easily connectedto surface or subsurface features e.g. the middle-oceanic ridge in the Atlantic and othersea-flour structures, Iceland’s hotspot signature, the Baltic sea depression and Spitzbergenvolcanic signal, the Canadian depression etc.

For the Moon the situation is less complicated because of the presumed end of dynamicprocesses in the lunar mantle. Hence all the phenomena on the Emax-kmax maps shouldhave an origin in observed topographical (e.g. craters, basins) or subsurface (mascons)features. Special attention should be given to the mascons (positive signals in the center ofdepressions – short for mass concentrations) as they are signs of complicated geologicalstructure initiated most likely by giant impacts. In Fig. 6.11 the mascons connected to MareImbrum, Mare Serenitatis, Mare Crisium and Mare Nectaris really dominate the near-sidewhereas on the far-side is the major feature the South Pole-Aitken basin with a negativesignal. However besides smaller craters and mare anomalies, one adjacent to the irregularMare Marginis appears (circled). This is very promising for new research as this area couldbe an old impact basin (Wilhelms, 1987) covered by subsequent tectonic activity. In Fig.6.12 only small-scale features mainly connected to the basin and crater rims are displayed– this offers a promising tool for localization of new mascons (Konopliv et al., 2001) inupcoming geoid models. The insufficient regularization of the used gravity model is obvioushere on the far-side.

The martian geoid is dominated by a bipolar structure created by the assumed single-cell mantle dynamics (e.g. Harder and Christensen, 1996) which has most probablyalready ceased. Therefore the long-wavelength analysis in Fig. 6.13 shows these two mainfeatures together with some smaller contributions from topographic structures like ElysiumMons, Isidis Planitia or Hellas basin. However in Fig. 6.14 we can observe a complicatedsignal surrounding Olympus Mons and Valles Marineris and also some sign-changing signal(circled) south of Elysium Mons. The latter is probably induced by the unusual topographicstructure observed here but some others connected to martian crust-dichotomy or ancientshore-line may be found through more detailed research on new gravity maps being obtainedby current space missions.


Figure 6.7: At top, the topography and geoid of Venus, at bottom the Emax and kmax mapsof the geoid for jw=08-16. We can clearly observe the dominant dynamic part of the geoidwhich coincides with major topographic masses


Figure 6.8: At top, the topography and geoid of Venus, at bottom the Emax and kmax mapsof the geoid for jw=32-64. Even with a small scale we can see good agreement between bothfields and coronae clustering in equatorial region between Atla and Beta Regiones


Figure 6.9: At top, the topography and geoid of Earth, at bottom the Emax and kmax mapsof the geoid for jw=08-16. In this case the contribution of dynamic processes is even moreevident than in the case of Venus because of the mismatch with the continental masses


Figure 6.10: At top, the topography and geoid of Earth, at bottom the Emax and kmax mapsof the geoid for jw=32-64. Most contributions can be assigned to the prominent surface oroceanic topographic or tectonic structures


Figure 6.11: At top, the topography and geoid of the Moon, at bottom the Emax and kmax

maps of the geoid for jw=08-16. We can observe mainly topographic and mascons’ signalhowever a prominent anomaly not directly connected to these could be found (circled)


Figure 6.12: At top, the topography and geoid of the Moon, at bottom the Emax andkmax maps of the geoid for jw=32-64. Here the mascons’ signal is dominant with clearlyobservable rims of mare basins and major craters


Figure 6.13: At top, the topography and geoid of Mars, at bottom the Emax and kmax mapsof the geoid for jw=08-16. Here the dominance by Tharsis and Utopia Planitia is obvioushowever some complex signal of other structures is observed


Figure 6.14: In upper part the topography and geoid of Mars, in lower one Emax and kmax

maps of geoid for jw=32-64. We can examine the signal of well known tectonic and impactfeatures together with some not yet well-known (circled)

Chapter 7

Discussion of Results

Our research was done on global planetary scale and thus results obtained by the usedinverse method should tell us important facts about general tendencies of Venus’ structure.However lack of any direct observational data from the venusian interior can put our findingsinto question – so an appropriate final step is to compare and discuss them with the resultsof forward models mainly performed previously by others.

Our analysis of the crustal structure is based on global data and so the individual siteexamination could differ slightly from a derived average compensation depth of 35 km.However even local admittance studies (Simons et al., 1997) are in agreement withour findings and predict for exclusive locations like Beta Regio, Niobe Planitia or Lak-shmi Planum very close values of crustal thickness. The earlier estimates of a global value(Arkani-Hamed, 1996) could differ slightly because of the use of older gravity and to-pography models. Also the localization of the observed and predicted geoid done in theprevious section (Fig. 6.2) confirms that, in the case of no longer active regions at Venus,the inferred compensation depth could well predict the observed data.

On the other hand for the long and intermediate wavelengths the dynamic flow model ofVenus’ mantle predicts a geoid and topography in very close agreement with the observedvalues as noted earlier by several authors (Kiefer et al., 1986; Phillips, 1990; Simonset al., 1997). So the inverse problem based on this technique could gives us importantanswers about the mantle rheology. The test of resolution of our inverse problem formulationconfirms that we can detect some major subsurface features, though with certain limits inobtaining the unambiguous values.

The three main results obtained (stiffer lithosphere overlaying the mantle, lack of thelow-viscosity zone – LVZ – and a gradual increase of viscosity through the mantle by order1-1.5 magnitude) are in agreement not only with papers dealing with Venus’ interior but alsowith some authors studying general states of convection in the case of systems with missingmobile plate tectonics. A viscosity contrast by at least 3 orders of magnitude from thelithosphere to the rest of the mantle and a lack of LVZ in a site of some major plumes wereshown by a regional modelling (Kiefer and Hager, 1991; Kiefer and Hager, 1992)and also indicated by some global convection studies (Dubuffet et al., 2000; Steinand Hansen, 2003). A common implication from these findings is that venusian rheology

71

CHAPTER 7. DISCUSSION OF RESULTS 72

is dryer, which is caused mainly by the early water escape (Donahue and Hodges, 1992)and therefore the plate tectonics could not evolve without an underlaying LVZ. Some othershowever (e.g. Simons et al., 1997), are assuming that a determination of the presenceof an LVZ on Venus is out of our current knowledge. The viscosity increase through themantle used to sometimes be neglected (Venus’s mantle is then handled as isoviscose) andsometimes interpreted as a upper/lower mantle boundary – but our results do not indicatesuch a sharp contrast, rather we explain it in terms of a monotonous viscosity rise.

The question of lithospheric thickness remains only partially answered because the dy-namic predictions are based on long wavelengths and are not very sensitive to shallowviscosity structures. But the basic result is between 100-200 km which corresponds alsowith an isostasy compensation model for the whole spectrum j=2-90 which gives ADCaround 165 km. This is in agreement with some of the estimates of lithospheric thickness(e.g. Nimmo and McKenzie, 1998) despite some other authors (e.g. Solomatov andMoresi, 1996) putting the bottom of the lithosphere at depths 200-400 km. It’s not clearif this parameter could be definitely derived from our present knowledge for example byregional modelling (see discussion in Nimmo and McKenzie, 1998) or if we have to waituntil some in-situ seismological measurements are done.

Also interesting for our investigation is the correspondence between the inducing den-sity perturbation power spectrum for wavelengths j=2-40 (Fig. 5.8 right) and observedtomography power spectrum (e.g. Cızkova et al., 1996). The linear tendency on thelogarithmic scale at intermediate wavelengths also appears in the averaged horizontal powerspectra of temperature from time-dependent convection of Newtonian fluid (Larsen etal., 1995). The similarity to our results brakes up around degree 40 (Fig. 5.8, right of thedashed line) which is also the interface between dynamic and isostatic theory use. Fromthat we can also conclude a the suitability of our dynamic flow model.

Chapter 8

Conclusions

In our search for a geoid generated primarily by dynamic processes in the mantle of Venuswe have to first separate the contribution of isostatic support of topography. For thisthe isostatic hypothesis seems to predict the observed data successfully and both inverseproblem formulations used (for degree apparent depth of compensation and fitting the degreeadmittance function) are steady and give the same results. However the obtained apparentdepth of compensation 35 km is only an averaged global approximation, the local crustalthickness can vary from that significantly.

Our work seems to exploit the theme of dynamic support of topography which hasalready been taken by many others. Nevertheless the unknown density distribution limitssuch models and they can answer only questions regarding basic mantle properties. Ourdensity approximation of only lateral variations constant with depth is one of the possibleways to overcome these difficulties. The inverse problem based on such a density model andinternal loading theory appears to be steady for small number of free parameters, as theirnumber increase, the solution gets less robust. Hence the major obtained results:

• the existence of a stiff lithosphere with a viscosity higher by several orders of magnitudethan the underlaying mantle

• the absence of a narrow low viscosity zone (similar to Earth’s asthenosphere) beneaththe lithosphere

• the gradual increase of viscosity through the mantle by 1-1.5 orders of magnitude

have an importance as a guiding solution showing the general trends in Venus’ mantle. All ofthese findings were already introduced by other authors but never before by means of a welldefined inverse problem and all together. Our contribution is therefore the confirmation ofthese results and the compilation of them into a coherent model. For better resolution ofthe rheology of the deep mantle another approach, following new data, would be necessary.

The use of localization methods appears to be another efficient tool in comparing andanalyzing spatial geophysical fields. For our purposes it has been the way to overcomethe disadvantages of the global character of the spherical harmonic approach and can beused for both field analysis and local-spectral studies. The Emax-kmax method combines

73

CHAPTER 8. CONCLUSIONS 74

both the spatial analytic and the synthetic approaches and despite certain mathematicaldisadvantages seems to be a valuable tool in the examination of planetary gravity fields.

As mentioned in the previous paragraph, our effort to determine the subsurface structureand mantle rheology is strongly limited by the unknown density distribution beneath thevenusian surface. The current knowledge of Venus’ geophysics, the collected data and theiranalysis cannot give us a satisfactory answer to this problem, and so we see the importanceof future space missions focused on detailed geophysical exploration for the sake of betterunderstanding planetary formation and evolution.

Appendix A

Spherical Harmonics

In many geophysical problems we need to describe spherical fields by means of scale-varyingglobal functions. A very useful tool is to employ spherical harmonic (SH) functions whichcreate an orthogonal basis on the sphere. Within this basis we can evaluate either scalar,vector or tensor fields to an appropriate set of coefficients and use it to express some PDEproblems as a set of ODE which are easier to solve.

A.1 Scalar Spherical Harmonics

Similar to Fourier decomposition of a continuous signal into a series, the spherical scalarfunction can be evaluated by the summation of coefficients multiplied by the appropriatebasis functions (e.g. Jones, 1985; Varshalovich et al., 1988):

f(ϑ, ϕ) =∞∑

j=0

j∑

m=−j

AjmYjm(ϑ, ϕ) (A.1)

for 2D function f or in the 3D case:

f(r, ϑ, ϕ) =∞∑

j=0

j∑

m=−j

Ajm(r)Yjm(ϑ, ϕ), (A.2)

where ϑ and ϕ are the colatitude and longitude respectively, Ajm and Ajm(r) the appropriateSH coefficients. The basis functions Yjm(ϑ, ϕ) are defined as follows:

Yjm(ϑ, ϕ) = (−1)mNjmPjm(cos ϑ)eimϕ j ≥ 0 m ≥ 0 (A.3)

Yjm(ϑ, ϕ) = (−1)mY ∗j|m|(ϑ, ϕ) j ≥ 0 m < 0, (A.4)

where the asterisk means complex conjugation, the normalization factor Njm is given by

Njm =[(2j + 1)

4π

(j −m)!

(j + m)!

] 12

(A.5)

75

APPENDIX A. SPHERICAL HARMONICS 76

and the Pjm in our definition of spherical harmonics are the associated Legendre functionsderived from Legendre polynomials Pj:

Pjm(cos ϑ) = (1− cos2 ϑ)m2

dmPj(cos ϑ)

d(cos ϑ)m= sinm ϑ

dmPj(cos ϑ)

d(cos ϑ)m(A.6)

Pj(x) =1

2jj!

dj

dxj(x2 − 1)j. (A.7)

Moreover if the function f ∈ C2 (i.e. the function has a continuous derivative) the seriesA.1 (resp. A.2) is uniformly convergent and therefore:

limj→∞

|fjm| = 0 (A.8)

The definitions of spherical harmonics used (A.3 and A.4) satisfy the orthonormalityconditions (the δ symbols stands for the Kronecker function δjm = 1 for the case j = m,otherwise it is equal to zero):

2π∫

0

π∫

0

Yjm(ϑ, ϕ)Y ∗kl(ϑ, ϕ) sin ϑdϑdϕ = δjkδml. (A.9)

To evaluate coefficients Ajm – similarly for Ajm(r) – we use the decomposition by a dotproduct of scalar field with the basis functions Yjm:

Ajm =

2π∫

0

π∫

0

f(ϑ, ϕ)Y ∗jm(ϑ, ϕ) sin ϑ dϑ dϕ (A.10)

In the case that f(ϑ, ϕ) is a real function we can evaluate Ajm, m < 0 using the symmetryfrom equation A.4 in relation A.10:

Aj,−m = (−1)mA∗jm. (A.11)

A.1.1 Applications

To calculate |fj|2 – the power spectra of SH coefficients – to estimate the contributionof different wavelength features we use the following expression or, in the case of a realfunction f , its symmetric modification according to equation A.11.

|fj|2 =j∑

m=−j

fjmf ∗jm. (A.12)

Another useful tool to investigate the global characteristic is a degree-correlation function cj,which gives the information about the similarity between two SH fields (e.g. between geoidgjm and topography tjm) at various scales:

cj =

j∑m=−j

gjmt∗jm√

j∑m=−j

gjmg∗jm

√j∑

m=−jtjmt∗jm

. (A.13)


If we want to evaluate a transfer degree-ratio between two SH fields we can use the ad-mittance function aj, which represents the part of the field correlated with the other one –whereas the uncorrelated part is represented by ujm (e.g. Schubert et al., 2001):

gjm = ajtjm + ujm (A.14)

aj =

j∑m=−j

tjmg∗jm

j∑m=−j

tjmt∗jm

. (A.15)

Considering Newton’s gravity law the gravity signal at point r of mass distributed insidethe sphere Ω′ could be evaluated in the form of the gravitational potential V (r):

V (r) = κ∫

Ω ′

ρ(r ′)|r − r ′| dΩ ′. (A.16)

In geophysics, rather than with the geopotential, we work with the geoid, though theseterms are nearly equivalent. The geoid height at any point on the surface located at (ϑ, ϕ)is then given (considering a local gravity acceleration g0) by the following equation:

g(ϑ, ϕ) =V (ϑ, ϕ)

g0(ϑ, ϕ). (A.17)

For our purposes we need to transform an equation A.16 into a SH form hence we must firsttransform the density ρ and the relative vector 1

|r−r ′| terms:

ρ(r ′) =∞∑

j=0

j∑

m=−j

ρjm(r ′)Yjm(ϑ ′, ϕ ′) (A.18)

1

|r − r ′| =4π

r

∞∑

k=0

k∑

l=−k

1

2k + 1

(r ′

r

)k

Y ∗kl(ϑ

′, ϕ ′)Ykl(ϑ, ϕ) , r ≥ r ′ (A.19)

which allow us to write equation A.16 in the form:

V (r) = κ∫

Ω ′

∑

j,m

∑

k,l

4π

r

1

2k + 1

(r ′

r

)k

ρjm(r ′)Yjm(ϑ ′, ϕ ′)Y ∗kl(ϑ

′, ϕ ′) ×

× Ykl(ϑ, ϕ)r ′ 2 sin ϑ ′dϑ ′dϕ ′dr ′. (A.20)

Using orthonormality relation A.9 we can simplify the previous equation into:

V (r) =4πκ

r

∞∑

j=0

j∑

m=−j

1

2j + 1

r2′∫

r1′

(r ′

r

)j

ρjm(r ′)Yjm(ϑ, ϕ)r ′ 2dr ′. (A.21)


When we want to apply the derived formula A.21 to the geopotential measured at thesurface of planet then r = R (the planetary radius) and we obtain:

V (r) =4πκ

R

∞∑

j=0

j∑

m=−j

1

2j + 1

[ R∫

0

(r ′

R

)j

ρjm(r ′)r ′ 2dr ′]Yjm(ϑ, ϕ). (A.22)

Because we want to get a SH series in the form

V (r) =∞∑

j=0

j∑

m=−j

Vjm(R)Yjm(ϑ, ϕ), (A.23)

we must define Vjm in the following form

Vjm(R) =4πκ

R

1

2j + 1

R∫

0

(r ′

R

)j

ρjm(r ′)r ′ 2dr ′. (A.24)

If we want to evaluate the geopotential contribution of density anomalies caused by topog-raphy undulations at given radius (r′ = R1) – e.g. at the surface or at MOHO – which couldbe evaluated as ρjm(r ′) = tjm∆ρ δR1 , where tjm are SH coefficients of this topography, ∆ρis the density contrast and δR1 = 0 other than at the given depth. Thus we obtain

Vjm(R) =4πκ

R

1

2j + 1

(R1

R

)j

tjm∆ρR21 =

4πκR

2j + 1

(R1

R

)j+2

tjm∆ρ (A.25)

Vjm(R) =4πκR

2j + 1tjm∆ρ, at the surface R1 = R. (A.26)

Dealing with a topography compensation we need to set the isostasy by the inverse to-pography sjm at the depth h with amplitudes proportional to density contrast ratio

∆ρsurf

∆ρMOHO:

sjm = −tjm∆ρsurf

∆ρMOHO

, (A.27)

therefore the gravitational potential from the compensated surface topography tjm (the sumof the equation A.25 for R1 = R, R− h) with the density contrast ∆ρ

.= ρsurf is:

Vjm =4πκR

2j + 1

(1−

((R− h)

R

)j+2)tjm∆ρ. (A.28)

If we want then to calculate the least-square misfit function between two SH fields (e.g.between observed and predicted geoid) we will use the following formula

S2 =jmax∑

j=0

j∑

m=−j

(gpredjm − gobs

jm)(gpredjm − gobs

jm)∗. (A.29)


However, for some geophysical purposes we may need to express a gravity field insteadin the form of a geoid in the form of free-air gravity. This is defined as a radial accelerationat the geoid level (usually measured in mGals):

a(R) =∂V (r)

∂r

∣∣∣∣r=R

. (A.30)

This equation is easy to transform into SH form by considering an appropriate definition ofthe Earth geopotential

V (r, ϑ, ϕ) =∞∑

j=0

j∑

m=−j

(R

r

)j+1

Vjm(R)Yjm(ϑ, ϕ). (A.31)

Putting A.31 relation into A.30 we obtain

Vjm(r) =(

R

r

)j+1

Vjm(R) (A.32)

∂Vjm(r)

∂r= (j + 1)

Rj+1

rj+2Vjm(R) (A.33)

ajm(R) =j + 1

RVjm. (A.34)

In the satellite geodesy there are conventional equations for SH expansion of the topog-raphy (A.35) and the geopotential (A.36) mostly set by Kaula, 1966:

T (ϑ, ϕ) = R∞∑

j=2

j∑

m=0

Pjm(cos ϕ)(Ctjm cos mϑ + St

jm sin mϑ) (A.35)

V (ϑ, ϕ) =GM

R+

GM

R

∞∑

j=2

j∑

m=0

Pjm(cos ϕ)(Cvjm cos mϑ + Sv

jm sin mϑ), (A.36)

where Ctjm, St

jm and Cvjm, Sv

jm are the corresponding fully normalized SH trigonometric co-efficients. Re-normalizing these trigonometric coefficients for un-normalized exponential SHcoefficients Ajm suitable for use in A.1 or A.2 is simple:

Atj0 = R

√4π(Ct

jm) Atjm = (−1)mR

√2π(Ct

jm + iStjm) (A.37)

Avj0 =

GM

R

√4π(Cv

jm) Avjm = (−1)m GM

R

√2π(Cv

jm + iSvjm), (A.38)

where GM, R and g0 are constants needed to evaluate either the topography or gravitationalfield – for the proper values of the parameters known today (those for Venus, Earth, Moonand Mars) see table 3.1. When using trigonometric coefficients it is common to write thepower spectra equation A.12 in a form (e.g. Arkani-Hamed, 1996)

RMSj =

√√√√∑j

m=0(C2jm + S2

jm)

2j + 1(A.39)


A.2 Vector Spherical Harmonics

When considering a vector spherical field we need a proper basis for its description. Insteadof using Cartesian (ex, ey, ez) or spherical (er, eϑ, eϕ) unit vectors we use so called cyclicunit vectors eµ (µ = −1, 0, 1), which are constructed in the following way

e1 = − 1√2(ex + iey) (A.40)

e0 = ez (A.41)

e−1 =1√2(ex − iey) (A.42)

and have these properties

e∗µ = (−1)µe−µ (A.43)

e∗µ · eµ′ = (−1)µe−µ · eµ′ = δµµ′ . (A.44)

The scalar spherical harmonic series from the previous section could be adapted to thevector case by adding one dimension to the used basis functions. Therefore the 2D or 3D(respectively) vector functions can be written in the form:

f (ϑ, ϕ) =∞∑

j=0

j∑

m=−j

j+1∑

l=|j−1|Al

jmY ljm(ϑ, ϕ) (A.45)

f (r, ϑ, ϕ) =∞∑

j=0

j∑

m=−j

j+1∑

l=|j−1|Al

jm(r)Y ljm(ϑ, ϕ), (A.46)

where Y ljm are the vector spherical harmonic functions. This vector canonic basis – which

is in fact a tensor basis A.58 for a case k=1 (this is omitted in the superscript, could bewritten also as Y l1

jm) – is defined in the following way (see e.g. Edmonds, 1960):

Y ljm(ϑ, ϕ) =

1∑

µ=−1

l∑

ν=−l

Cjmlν1µYlν(ϑ, ϕ)eµ (A.47)

where Cjmlν1µ are Clebsh-Gordan coefficients (e.g. Varshalovich et al., 1988). Alterna-

tively the vector spherical harmonic functions could be defined by use of the spherical basis(er, eϑ, eϕ):

√j(2j + 1)Y j−1

jm (ϑ, ϕ) = jYjm(ϑ, ϕ)er +∂Yjm(ϑ, ϕ)

∂ϑeϑ +

1

sin ϑ

∂Yjm(ϑ, ϕ)

∂ϕeϕ

(A.48)√

j(j + 1)Y jjm(ϑ, ϕ) = i

1

sin ϑ

∂Yjm(ϑ, ϕ)

∂ϕeϑ − i

∂Yjm(ϑ, ϕ)

∂ϑeϕ (A.49)

√(j + 1)(2j + 1)Y j+1

jm (ϑ, ϕ) = −(j + 1)Yjm(ϑ, ϕ)er +

+∂Yjm(ϑ, ϕ)

∂ϑeϑ +

1

sin ϑ

∂Yjm(ϑ, ϕ)

∂ϕeϕ. (A.50)


The appropriate SH coefficients f ljm (resp. f l

jm(r)) can be derived by a modification ofequation A.10 in the sense of term A.45:

f ljm =

∫ π

0

∫ 2π

0f (ϑ, ϕ) ·Y l ∗

jm(ϑ, ϕ) sin ϑ dϑdϕ. (A.51)

Because of the symmetries of Clebsh-Gordan coefficients we can again evaluate both Y lj,−m

and therefore even f lj,−m by using the terms with m ≥ 0:

Y lj,−m = (−1)j+m+l+1Y l ∗

jm (A.52)

f lj,−m = (−1)j+m+l+1f l ∗

jm (A.53)

Such basis functions defined by term A.47 (resp. by equations A.48-A.50) satisfy thecondition of orthonormality:

∫ π

0

∫ 2π

0Y l1

j1m1(ϑ, ϕ) ·Y l2 ∗

j2m2(ϑ, ϕ) sin ϑ dϑdϕ = δj1j2δm1m2δl1l2 . (A.54)

Moreover it could be shown that the field on the sphere can be separated into a toroidal partwhich describes only horizontal changes (∇.vT = 0 and vT .er = 0) and a spheroidal part((∇.× vS).er = 0 – when we deal with the non-divergence field i.e. ∇.v = 0 then this partis called poloidal). The f j±1

jm coefficients then represent the spheroidal and f jjm the toroidal

part of the field f (e.g. Matas, 1995).

A.3 Tensor Spherical Harmonics

If we have a spherical field of tensor quantity we can adapt the cyclic basis (A.40-A.42) toobtain a tensor orthogonal one:

E kλ =1∑

µ=−1

1∑

ν=−1

Ckλ1µ1νeµeν . (A.55)

With an appropriate basis function we can again write a series for SH expansion of aspherical field – this time the 2D or 3D tensor one:

F (ϑ, ϕ) =∞∑

j=0

j∑

m=−j

j+1∑

l=|j−1|

2∑

k=0

AlkjmY lk

jm(ϑ, ϕ) (A.56)

F (r, ϑ, ϕ) =∞∑

j=0

j∑

m=−j

j+1∑

l=|j−1|

2∑

k=0

Alkjm(r)Y lk

jm(ϑ, ϕ) (A.57)

where the spherical harmonic functions Y kljm are defined (e.g. Varshalovich et al.,

1988) by:

Y lkjm(ϑ, ϕ) =

l∑

µ=−l

k∑

ν=−k

CjmlνkµYlν(ϑ, ϕ)E kµ, (A.58)


and again satisfy the orthogonality relation:

∫ π

0

∫ 2π

0Y l1k1

j1m1(ϑ, ϕ) : Y l2k2 ∗

j2m2(ϑ, ϕ) sin ϑ dϑdϕ = δj1j2δm1m2δl1l2δk1k2 , (A.59)

where : operator denotes the double-dot product (for second order tensors with componentsAij and Bij is defined as A : B =

∑ ∑AijBij). Moreover such an orthonormal basis has

a straight-forward interpretation: the coefficients F l0jm represents the trace of the tensor

function, F l1jm stands for the antisymmetric and F l2

jm for the symmetric deviatoric part ofthe tensor function F . The evaluation of these coefficients is again only a modification ofprevious relations A.10 and A.51:

f lkjm =

∫ π

0

∫ 2π

0F (ϑ, ϕ) : Y lk ∗

jm (ϑ, ϕ) sin ϑ dϑdϕ. (A.60)

A.4 Operations with Spherical Harmonics

In this section various products of spherical harmonics and others operations are listed –some of them are used in Appendix C, others could be useful in dealing with differentialequations in the form of SH expansions. Most of the formulae are from Varshalovich etal., 1988; Matas, 1995 and the rest came from Cadek, personal communication.

Products of spherical harmonic

Yj1m1Yj2m2 =Πj1j2√

4π

∑

jm

1

Πj

Cj0j10j20C

jmj1m1j2m2

Yjm

Yj1m1Yl2j2m2

=Πj1j2l2√

4π

∑

jml

(−1)j+l+1 C l0j10 l20 Cjm

j1m1 j2m2

j j1 j2

l2 1 l

Y l

jm

Y l1j1m1

·Y l2j2m2

= (−1)j2+l2Πj1j2l1l2√

4π

∑

jm

1

Πj

Cj0l10 l20 Cjm

j1m1 j2m2

j1 j2 j

l2 l1 1

Yjm

Y l1j1m1

×Y l2j2m2

=i√

3√2π

Πj1j2l1l2

∑

jml

C l0l10 l20 Cjm

j1m1 j2m2

j j1 j2

l l1 l2

1 1 1

Y ljm

Y l1j1m1

Y l2j2m2

=Πj1j2l1l2√

4π

∑

jmln

Πn C l0l10 l20 Cjm

j1m1 j2m2

j l n

jl l1 1

j2 l2 1

Y lnjm

Y l1n1j1m1

: Y l2n2j2m2

= δn1n2(−1)j2+l2Πj1j2l1l2√

4π

∑

jm

1

Πj

Cj0l10 l20 Cjm

j1m1 j2m2

n1 l1 j1

j j2 l2

Yjm


Y l1j1m1

·Y l2n2j2m2

= (−1)n2+1 Πj1j2l1l2n2√4π

∑

jml

C l0l10 l20 Cjm

j1m1 j2m2

j j1 j2

l l1 l2

1 1 n2

Y ljm

Y l1n1j1m1

·Y l2j2m2

= (−1)n1+1 Πj1j2l1l2n1√4π

∑

jml

C l0l10 l20 Cjm

j1m1 j2m2

j j1 j2

l l1 l2

1 n1 1

Y ljm

where the Πj1j2...jn =n∏

i=1

√2ji + 1, the term j1 j2 j

l1 l2 l stands for 6-j Wigner symbol whereas

the similar one constituted of 3 lines for 9-j Wignal symbol. The definitions of these symbolsare given e.g. in Varshalovich et al., 1988 and their properties are handled later inthis section.

Formulae for products of a unit radial vector and spherical tensors

erYjm =1√

2j + 1(√

j δl,j−1 −√

j + 1 δl,j+1) Y ljm

er ·Y ljm =

1√2j + 1

(√

j δl,j−1 −√

j + 1 δl,j+1) Yjm

er ·Y lnjm = (−1)j+l

√2n + 1

√l + 1

l n j

1 l + 1 1

Y l+1

jm −√

l

l n j

1 l − 1 1

Y l−1

jm

er ·Y j0jm =

1√3(2j + 1)

(√

j + 1 Y j+1jm −

√j Y j−1

jm )

er ·Y j−2,2jm =

√j − 1

2j − 1Y j−1

jm

er ·Y j−1,2jm =

√j − 1

2(2j + 1)Y j

jm

er ·Y j,2jm =

√√√√ j(2j − 1)

2.3.(2j + 1)(2j + 3)Y j+1

jm −√√√√ (j + 1)(2j + 3)

2.3.(2j + 1)(2j − 1)Y j−1

jm

er ·Y j+1,2jm = −

√j + 2

2(2j + 1)Y j

jm

er ·Y j+2,2jm = −

√j + 2

2j + 3Y j+1

jm


(v · er)er =∑

jm

1√2j + 1

×[

jvj−1jm −

√j(j + 1)vj+1

jm

]Y j−1

jm −[√

j(j + 1)vj−1jm − (j + 1)vj+1

jm

]Y j+1

jm

v − (v · er)er =∑

jm

vjjmY j

jm+

∑

jm

1

2j + 1

[(j + 1)vj−1

jm +√

j(j + 1)vj+1jm

]Y j−1

jm +[√

j(j + 1)vj−1jm + jvj+1

jm

]Y j+1

jm

Differential operators acting on spherical tensors

∆ [f(r)Yjm] =

[d2f(r)

dr2+

2

r

df(r)

dr− j(j + 1)f(r)

r2

]Yjm

∇ [f(r)Yjm] =1√

2j + 1

[√j

(d

dr+

j + 1

r

)f(r)Y j−1

jm −√

j + 1

(d

dr− j

r

)f(r)Y j+1

jm

]

∇[f(r)Y l

jm

]= (−1)j+l+1

∑

k

√2k + 1

√l

1 1 k

j l − 1 l

(d

dr+

l + 1

r

)f(r)Y l−1,k

jm +

+(−1)j+l∑

k

√2k + 1

√l + 1

1 1 k

j l + 1 l

(d

dr− l

r

)f(r)Y l+1,k

jm

∇ · f(r)Y ljm =

1√2j + 1

[√j

(d

dr− j − 1

r

)δl,j−1 −

√j + 1

(d

dr+

j + 2

r

)δl,j+1

]f(r)Yjm

∇× f(r)Y ljm = −i

√6(−1)j+l

√l + 1

l j 1

1 1 l + 1

(l

r− d

dr

)f(r)Y l+1

jm−

−i√

6(−1)j+l√

l

l j 1

1 1 l − 1

(l + 1

r+

d

dr

)f(r)Y l−1

jm

∆[f(r)Y l

jm

]=

[d2f(r)

dr2+

2

r

df(r)

dr− l(l + 1)f(r)

r2

]Y l

jm

∆[f(r)Y ln

jm

]=

[d2f(r)

dr2+

2

r

df(r)

dr− l(l + 1)f(r)

r2

]Y ln

jm


∇ ·[f(r)Y lk

jm

]= (−1)j+l

√2k + 1 ×

√l + 1

1 l l + 1

j 1 k

(d

dr− l

r

)f(r)Y l+1

jm −√

l

1 l l − 1

j 1 k

(d

dr+

l + 1

r

)f(r)Y l−1

jm

Clebsch-Gordan coefficients

More on the definition of Clebsch-Gordan coefficients and related Wigner symbols couldbe found again in Varshalovich et al., 1988 or [i8]. Here we will be dealing just withthe properties most useful to our purpose.

¦ Symmetries of Clebsch-Gordan coefficients

Cjmj1m1 j2m2

= (−1)j+j1+j2 Cj,−mj1,−m1 j2,−m2

= (−1)j+j1+j2 Cjmj2m2 j1m1

Cjmj1m1 j2m2

= (−1)j1+m1Πj

Πj2

Cj2,−m2j1m1 jm

Cjmj1m1 j2m2

= (−1)j1+m1Πj

Πj2

Cj2m2jm j1,−m1

Cjmj1m1 j2m2

= (−1)j2+m2Πj

Πj1

Cj1,−m1j,−m j2m2

Cjmj1m1 j2m2

= (−1)j2+m2Πj

Πj1

Cj1m1j2,−m2 jm

¦ Values of coefficients for a special choice of indices

Cj0j10 j20 = 0 if j1 + j2 + j3 is odd

Cj+1,0j0 10 =

√j + 1

2j + 1

Cj−1,0j0 10 = −

√j

2j + 1

Cjmj1m1 00 = δjj1δmm1

C00j1m1 j2m2

=(−1)j1+m1

Πj1

δj1j2δm1,−m2

Cjjj1j1 j2j2 = δj1+j2,j


¦ Sums of Clebsch-Gordan coefficients

∑m1m2

Cjmj1m1 j2m2

Cj′m′j1m1 j2m2

= δjj′δmm′

∑mm1

Cjmj1m1 j2m2

Cjmj1m1 j3m3

=Πj

Πj2

δj2j3δm2m3

∑n1n2n3

Cj3m3

l1n1 l2n2Cj1m1

l3n3 l2n2C l3n3

l1n1 j2m2= (−1)j1+l2+l3 Πj3l3 Cj1m1

j2m2 j3m3

l1 l2 j3

j1 j2 l3

∑n1n2n3n4

Cj1m1

l1n1 l2n2Cj2m2

l3n3 l4n4Cj3m3

l3n3 l1n1Cj4m4

l4n4 l2n2=

(−1)j+j1+j2 Πj1j2j3j4

∑

jm

Cjmj1m1 j2m2

Cjmj3m3 j4m4

l2 l1 j1

l4 l3 j2

j4 j3 j

¦ Clebsch-Gordan coefficients and 3-j Wigner symbols

Cjmj1m1 j2m2

= (−1)j1+j2+m Πj

j1 j2 j

m1 m2 −m

j1 j2 j

m1 m2 −m

=

(−1)j+m

Πj

Cjmj1,−m1 j2,−m2

6-j Wigner symbols

¦ Values of 6-j symbols for a special choice of indices

j1 j2 j

l1 l2 0

=

(−1)j1+j2+j

Πj1j2

δj1l2δj2l1

j 2 j − 2

1 j − 1 1

=

1√5(2j − 1)

j 2 j + 2

1 j + 1 1

=

1√5(2j + 3)

j j 2

1 1 j − 1

=

√√√√ (j + 1)(2j + 3)

2.3.5.j(2j − 1)(2j + 1)


j j 2

1 1 j + 1

=

√√√√ j(2j − 1)

2.3.5.(2j + 3)(j + 1)(2j + 1)

j 1 j

2 j + 1 1

= −

√j + 2

2.5.(j + 1)(2j + 1)

j 1 j

2 j − 1 1

= −

√j − 1

2.5.j(2j + 1)

j j 1

1 1 j + 1

=

√j

2.3.(j + 1)(2j + 1)

j j 1

1 1 j − 1

= −

√j + 1

2.3.j(2j + 1)

9-j Wigner symbols

¦ Values of 9-j symbols for a special choice of indices

0 j j

l k1 k2

l k3 k4

=(−1)j+l+k2+k3

Πjl

k4 k2 j

k1 k3 l

Appendix B

Spectral-spatial Methods

Frequent geophysical task is to display features of physical fields but only at a certain scale.For this purpose harmonic analysis could be used when the degree corresponds to a desiredsize. But this approach is complicated by the global character of harmonic functions – sincethe value at any place in the reconstructed field (e.g. topography of the required wavelengthat the north pole) is influenced by the rest of the field’s contributions to a given sphericalharmonic coefficient.

B.1 Continuous Wavelet Transform

Another possible way to solve this problem is to employ a continuous wavelet transform,which a method is widely discussed by Daubechies, 1992; Torrence and Compo,1998; Vecsey, 2002 et al. This method is based on the convolution of the wavelet functionψ0(a) (where a is a desired scale) – which is well localized in both the space and spectrum– with the given physical field, and enables us to display just the local anomalies of thesize corresponding to the scale of the wavelet. For the wavelet we can use any odd (betterresolution of edges) or even (better space resolution) function (Vecsey, 2002) but it mustsatisfy the condition of admissibility and unit energy of the whole space:

∞∫

−∞ψ0(η)dη = 0 (B.1)

∞∫

−∞|ψ0(η)|2dη = 1, (B.2)

where equation B.1 keeps the wavelet transform unaffected by the mean of analyzed functionand equation B.2 provides the normalization of resulting wavelet transform. For our pur-poses we use the so-called Mexican Hat (second derivative of the Gaussian) as the waveletfunction for both 1D and 2D. This wavelet is isotropic – and therefore detects the featuresfrom all directions with the same width and weight – and well localized in space which isnecessary to obtain the good space resolution.

89

APPENDIX B. SPECTRAL-SPATIAL METHODS 90

For displaying results (scalograms) we need an additional dimension for scale – this isnot a problem in the case of 1D but for the 2D wavelet transform we have to choose between3D diagrams (which are difficult to display) and slices showing one certain scale. In thiswork we use the second method and show several slices for certain distinguished scales.

B.1.1 1D Wavelets

In the case of 1D signal analysis we can use the wavelet transform (B.3) to display con-tributions of different frequencies to the original data. This method can be a useful toolfor such things as a time-series study (Vecsey and Matyska, 2001) or raw-data anal-ysis (Turcotte et al., 2002). For displaying results the time/period scalograms or asuperposition of wavelet transform with the gradual offset are suitable. When dealing witha Cartesian signal we can separate high-significance data from the white or red noise byusing a cone of influence (Torrence and Compo, 1998; Vecsey, 2002). Below is thedefinition of the 1D wavelet transform and its spectrum:

Ψf (a, b) = a−12

∞∫

−∞f(x)ψ∗0(

x− b

a) dx (B.3)

Ψf (a, ω) = a12 f(x)ψ∗0(aω), (B.4)

where b is the position of the wavelet transform, f(x) is the analyzed function, asteriskdenotes the complex conjunction, hat the Fourier transform and ω is the frequency inFourier space. Because of the homogeneity of the signal f(x) we can convert the wavelettransform (B.3) into a Fourier space (B.4) and vice versa – this is a very efficient way ofcomputing as it rapidly decreases the computational time.

B.1.2 2D Wavelets

For a 2D wavelet transform (B.5) on the Cartesian domain we use a transformation similarto the 1D case but enhanced to another dimension. Such an extension could be carried outeven for more dimensions (e.g. studying 3D structures – see Bergeron et al., 1999)but then the computational requirements increase and displaying the results becomes moredifficult. As well as in the previous case a transform into Fourier space (B.6) is again veryfavorable and spares a lot of computational time. The definition of the 2D wavelet transformand its spectrum is as follows:

Ψf (a, b) = a−1

Lx∫

0

Ly∫

0

f(x )ψ∗0(x − b

a) dx 2 (B.5)

Ψf (a, k) = af(k)ψ∗0(ak), (B.6)


Figure B.1: 2D Mexican Hat wavelet function

where k is the wavevector in the Fourier space. An often used 2D wavelet is the MexicanHat (Fig. B.1), which is defined on the Cartesian domain as a second derivative of theGaussian normal function centered at the origin and dependant only on the radial vectorr = r(x, y):

ψ0(r) =1√2π

(∂2

∂x2+

∂2

∂y2)e−

12|r |2 =

1√2π

(2− |r |2)e− 12|r |2 (B.7)

B.2 2D Spherical Localization

When we apply the method of a 2D wavelet transform to data fields on the sphere (e.g.gravity, topography etc.) transformed by a cylindrical projection to Cartesian coordinates,only regional studies should be considered. But when we need to analyze whole sphericaldata sets we obtain a distortion increasing with latitude (e.g. Vecsey et al., 2003).A solution to this problem is to substitute a wavelet transform by another transformationdefined on a sphere. It employs wavelets or other localized functions with a spherical cor-rection (Fig. B.2) which in the Euclidian limit gives the 2D wavelet transform as mentionedin previous section. Several different approaches have been suggested (Antoine et al.,2002; Michel, 2002) but they are usually very mathematically complicated and are rarelyemployed by geophysical community. Therefore some easier methods have been proposed –here two which are widely used in the planetary sciences are outlined.


Figure B.2: 2D Mexican Hat with the spherical correction

B.2.1 Spectral Approach

The general principle of the localization is (as well as in the case of the wavelet transform)based on the convolution of an analyzed field A(Ω) with a window function W (Ω) on thesphere Ω. The first possible method was presented by Simons et al., 1997 and it is real-ized purely in the spectral domain by operations with spherical harmonic (SH) coefficients(equations B.8-B.11). However one must remember that the following sequence of obtain-ing the localized field is only for a certain window-scale and window-position – for anotherscale or position of the window function on the sphere some other set of SH coefficient isobtained. In the following terms the normalization and phase convention are the same asin e.g. Varshalovich et al., 1988:

Ψ(Ω) = W (Ω)A(Ω) =∑

jm

ψjmYjm(Ω) (B.8)

ψjm =∫

A(Ω)W (Ω)Y ∗jm(Ω) dΩ (B.9)

ψjm =∑

j1m1j2m2

a∗j1m1w∗

j2m2ξj1j2l

j1 j2 j

0 0 0

j1 j2 j

m1 m2 m

(B.10)

ξj1j2...jn =

√(2j1 + 1)(2j2 + 1) . . . (2jn + 1)

4π(B.11)

where ajm, wjm and ψjm are respectively SH coefficients of analyzed field, window functionand obtained localized field; (j1j2j

0 0 0) and (j1 j2 jm1m2m) are the Wigner 3-j symbols defined e.g. by

Varshalovich et al., 1988. Using the ψjm and γjm coefficients of two localized fieldswe can construct the cross-covariance coefficient (B.12), RMS amplitude of Ψ (B.13), thecorrelation between Ψ and Γ (B.14) and the transfer function (admittance) (B.15):


σ2ΨΓ(j) =

∑m

ψjmγ∗jm (B.12)

Sj(Ω) =

√√√√σ2ΨΨ(Ω)

2j + 1(B.13)

rj(Ω) =σ2

ΨΓ(Ω)√σ2

ΨΨ(Ω)σ2ΓΓ(Ω)

(B.14)

Fj(Ω) =σ2

ΨΓ(Ω)

σ2ΨΨ(Ω)

. (B.15)

B.2.2 Spatial Approach

Another approach to a spherical localization is based on spatial computation and was pre-sented by Kido et al., 2003. The main advantages of this method are its computationalsimplicity and an option to process only a selected interval instead of the whole sphericalsurface, which is essential when we desire to work with high resolution or short wavelengths.Therefore we choose this system (B.16-B.17) for our calculations in Chapter 6 with parame-ters used by the authors. As a window function we apply the spherical wavelet-like functionFjw,σ,ϑ0,ϕ0(ϑ, ϕ) centered in (ϑ0, ϕ0) with wavelength jw and optional parameter σ, whichcontrols the width of the wavelet-like function:

Fjw,σ,ϑ0,ϕ0(ϑ, ϕ) = j2w exp

(−

(jwξ

2σ

)2)[J0(jwξ)

ξ

sin ξ− exp(−σ2)

](B.16)

cos ξ = cos ϑ0 cos ϑ + sin ϑ0 sin ϑ cos(φ0 − φ). (B.17)

where ξ is the angular distance to the center of the wavelet-like function (ϑ0, ϕ0) and J0 isthe Bessel function of order zero (the resulting function is very close to Mexican hat waveletmentioned in the previous section). Using these definitions we can evaluate the transformedfield A (B.18), the correlation between two fields (in our case G and T i.e. geoid andtopography respectively) (B.19) and the admittance between these fields (B.20) – both ofwhich are using the appropriate window (B.21) which also acts as a weighting function ifwe understand this problem as a least square fit.

Ajw,σ(ϑ0, ϕ0) =1

4π

2π∫

0

π∫

0

A(ϑ, ϕ)Fjw,σ,ϑ0,ϕ0(ϑ, ϕ) sin ϑ dϑ dϕ (B.18)

Cjw,σ(ϑ0, ϕ0) =

∫Ω Wjw,σ(ξ)T (ϕ, ϑ)G(ϕ, ϑ)dΩ√∫

Ω Wjw,σ(ξ)T 2(ϕ, ϑ) dΩ∫Ω Wjw,σ(ξ)G2(ϕ, ϑ) dΩ

(B.19)

Zjw,σ(ϑ0, ϕ0) =

∫Ω Wjw,σ(ξ)T (ϕ, ϑ)G(ϕ, ϑ)dΩ

∫Ω Wjw,σ(ξ)T 2(ϕ, ϑ) dΩ

(B.20)


Wjw,σ(ξ) = exp

−

(jwξ

2σ

)2 (B.21)

The localized geoid and topography analysis are progressive tools in looking for thelocal geophysical mechanisms (e.g. crustal thickness by Simons et al., 1997 or elasticlithosphere thickness by McGovern et al., 2002) whereas the spatial correlation andadmittance maps can examine a lateral variation of geoid/topography relations (e.g. esti-mations of different generating mechanisms by Kido et al., 2003). They can also helpus in the future to improve new spherical harmonic models of the geoid which would be thebasis for a better determination of the topographical elevation (Konopliv et al., 1999and Rappaport et al., 1999).

B.3 Compositional Methods

One of the disadvantages in the localization of geophysical fields is the difficulty displayingthe 2D analysis results and the redundant information contained in CWT. Hence a methodcombining the advantages of a localization approach with synthetic processing is needed –in this thesis we employed a modification of the Emax-kmax method (Bergeron et al.,1999) for spherical fields:

Emax(ϑ, ϕ) = max〈jw1,jw2〉

|ψjw,σ(ϑ, ϕ)|2 (B.22)

Emax(ϑ, ϕ) = max〈jw1,jw2〉

|ψjw,σ(ϑ, ϕ)|2 · sgn ψjw,σ(ϑ, ϕ). (B.23)

This technique simply analyzes the observed data on a given interval 〈jw1, jw2〉 by meansof localization and registers the highest value of the transformed field for each grid point(Emax B.22 or Emax B.23, if we take into an account the sign of signal) and the correspondingwavelength jw – these quantities are then displayed in new 2D maps.

Appendix C

Dynamic Modelling

The aim of dynamic modelling in our work is to predict a rheology of the venusian mantle.Because we do not have any information about the density structure, phase transitions,layering nor geochemical composition we have to find an adequate method which can in-corporate the only known quantities – gravitational potential and topography. Thereforewe will try to predict the observed data by means of the internal loading theory which wasalready successfully used earlier for research on Earth (e.g. Cadek et al., 1997) and alsoon Venus (e.g. Kiefer and Peterson, 2003).

C.1 Governing Equations

The mantle flow in our model is controlled only by the density structure of the venu-sian mantle (which controls the buoyancy forces) and its viscous profile. The basic set ofequations must then incorporate the continuity equation, the equation of motion and therheology equation. For our purpose we consider several simplifications (neglecting of theself-gravitation and compressibility of newtonian material) but despite that we can still usethis model to predict the venusian geoid and topography with high accuracy (Kiefer etal., 1986). Hence the equations of flow are handled in the following form:

∇ · τ + ρ g = 0 (C.1)

∇ · v = 0 (C.2)

τ = −I p + η(∇v +∇Tv), (C.3)

where τ is the stress tensor, ρ stands for the density, g for the normal gravity acceleration, vfor the flow velocity, p for the hydrostatic pressure and η for the viscosity. However becausethe internal structure of Venus is not known we keep g constant trough the whole model.The geoid and topography induced by the flow described by equations C.1-C.3 does notgenerally depend on the absolute values of viscosity and therefore our results can revealonly relative viscosity contrasts.

95

APPENDIX C. DYNAMIC MODELLING 96

C.2 Parametrization and Numerical Solution

The governing equations are solved for the whole mantle, i.e. in the spherical layer betweensurface and core-mantle boundary. At both boundaries we employ the free slip conditionC.4 (because of zero tangential stress) and zero vertical velocity C.5 (because no materialis penetrating trough the surface nor CMB) as reasoned by e.g. Kyvalova, 1994:

v · er = 0 (C.4)

τ · er − [(τ · er) · er] er = 0. (C.5)

Now we have to choose the numerical method to work with these mathematical terms. Asshown in Appendix A, a very reasonable way to do that is to employ the spherical harmonics.By using them we can express all the scalar (A.2), vector (A.46) and tensor (A.57) quantitiesby terms containing only appropriate coefficients and harmonic functions. Because the termsare not coupled together through different orders we can separate them into individualequations for every degree and order. Thus we can rewrite boundary conditions (C.4 andC.5) into:

√jvj−1

jm (r)−√

j + 1vj+1jm (r) = 0 (C.6)

j + 1

2j + 1

√j − 1

2j − 1τ j−2,2jm (r)− 1

2j + 1

√j(j + 1)(j + 2)

2j + 3τ j+2,2jm (r) −

−√√√√ 3(j + 1)

2(2j − 1)(2j + 1)(2j + 3)τ j2jm(r) = 0, (C.7)

the equation of motion (C.1) into a set of two equations:

−√

j

3(2j + 1)

(d

dr+

j + 1

r

)τ j0jm(r) +

√j − 1

2j − 1

(d

dr− j − 2

r

)τ j−2,2jm (r) − (C.8)

−√√√√ (j + 1)(2j + 3)

6(2j − 1)(2j + 1)

(d

dr+

j + 1

r

)τ j2jm(r) = −f j−1

jm (r)

√j + 1

3(2j + 1)

(d

dr− j

r

)τ j0jm(r) +

√√√√ j(2j − 1)

6(2j + 3)(2j + 1)

(d

dr− j

r

)τ j2jm(r) − (C.9)

−√

j + 2

2j + 3

(d

dr+

j + 3

r

)τ j+2,2jm (r) = −f j+1

jm (r),

the continuity equation (C.2) into:

√j

2j + 1

(d

dr− j − 1

r

)vj−1

jm (r)−√

j + 1

2j + 1

(d

dr+

j + 2

r

)vj+1

jm (r) = 0, (C.10)


and the constitutive relation (C.3) into three equations:

τ j−2,2jm (r)− 2η

√j − 1

2j − 1

(d

dr+

j

r

)vj−1

jm (r) = 0 (C.11)

τ j2jm(r) + 2η

√√√√ (j + 1)(2j + 3)

6(2j − 1)(2j + 1)

(d

dr− j − 1

r

)vj−1

jm (r) −

−2η

√√√√ j(2j − 1)

6(2j + 3)(2j + 1)

(d

dr+

j + 2

r

)vj+1

jm (r) = 0 (C.12)

τ j+2,2jm (r) + 2η

√j + 2

2j + 3

(d

dr− j + 1

r

)vj+1

jm (r) = 0. (C.13)

However these terms describe only the spheroidal part of the governing equations – thereis no toroidal flow in the mantle with only radial viscosity structure (e.g. Ricard andVigni, 1989). Therefore the corresponding terms vj

jm, τ j−1,2jm and τ j+1,2

jm are identically

equal to zero. Now we have a set of 6 equations for 6 unknown quantities vj−1jm (r), vj+1

jm (r),

τ j−2,2jm (r), τ j2

jm(r), τ j0jm(r) and τ j+2,2

jm (r) for each degree and order. We will solve this problemfor j ≥ 2 because we are interested mainly in the explanation of observed local geoid andtopographic features rather than in the center-of-mass/center-of-figure offset. In equationsC.8 and C.9 the terms f j−2,2

jm and f j−2,2jm denote the SH coefficients of buoyancy force:

f j−1jm = −gρjm

√j

2j + 1f j+1

jm = gρjm

√j + 1

2j + 1. (C.14)

If we have a solution of this set of equations we can easily calculate the amplitudes of thedynamic topography t. When we start with the term for hydrostatic equilibrium at thetop/bottom boundary (− and + sign respectively) we quickly obtain the result:

∓(τ · er) · er = ∆ρg0t ⇒ t = ∓(τ · er) · er

∆ρg0

. (C.15)

Transforming this result into the spherical harmonic form gives:

tjm(r) = ∓ 1

∆ρg0

(1√3τ j 0jm(r)−

√√√√ j(j − 1)

(2j + 1)(2j − 1)τ j−2,2jm −

−√√√√ (j + 1)(j + 2)

(2j + 1)(2j + 3)τ j+2,2jm +

√√√√ 2j(j + 1)

3(2j − 1)(2j + 3)τ j,2jm

). (C.16)

The dynamically generated geopotential is then generated by contributions from surfaceand CMB topography (we can use terms A.26 and A.25 respectively) and from the unevenmass distribution in the mantle:

Vjm = V surfjm + V mantle

jm + V CMBjm (C.17)

V mantlejm =

4πκR

2j + 1

Rsurf∫

RCMB

(r ′

R

)j+2

ρjm(r ′) dr ′ (C.18)


core

surface

RH, CE, BC

EM

RH, CE

RH, CE

RH, CE, BC

EM

r1

r2

rn-1

rn

s1

sn-1

t1 1, v

t2

v2

vn-1

tn

tn+1 n, v

Figure C.1: The alternating scheme for finite difference method – at every layer boundarywe calculate τ and at every center of layer v by using the appropriate equations. RH in thedescription stands for rheological equation (C.3), CE for continuity equations (C.2), BC forboundary conditions (C.4, C.5) and EM for the equation of motion (C.1)

To solve all these differential equations we use the spherical harmonic approach for ϑ, ϕcoordinates and for the r coordinate we employ the method of finite differences. We dividethe mantle into n− 1 layers and the derivatives in these equations then become:

df

dr(r1) ' 1

h(f1 − f2) (C.19)

df

dr(ri) ' 1

2h(fi−1 − fi) i = 2, . . . , n− 1 (C.20)

df

dr(rn) ' 1

h(fn−1 − fn), (C.21)

where h is the semi-thickness of the particular layer (from its boundary to its center). Thecomputational grid does not have all quantities defined at common layer boundaries but werather use the alternating scheme because of its numerical stability (Kyvalova, 1994).This defines the components of stress tensor τ at layer boundaries, whereas the componentsof velocity v are evaluated at their centers. At the uppermost and lowermost boundaries wegroup both τ and v together because the boundary conditions need them to be evaluated(see Fig. C.1).

When we divide the mantle into n−1 layers we have n boundaries but at the surface andCMB we also need to evaluate boundary conditions. Therefore we have 6n + 2 equationsand by appropriate positioning we can obtain the band-matrix which is easy to solve e.g.by FORTRAN numerical libraries (Numerical Recipes etc.). The response functions forconstant viscosity trough the mantle (Fig. C.2) show the contributions of load correspondingto the layer thickness at every single layer for both gravitational potential and topography.


-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

-0.0

geoid

[m

]

0 1000 2000 3000

depth [km]

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

surf

ace topogra

phy [m

]

0 1000 2000 3000

depth [km]

-6

-5

-4

-3

-2

-1

0

CM

B topogra

phy [m

]

0 1000 2000 3000

depth [km]

degree 2

degree 4

degree 8

degree 16

degree 32

degree 2

degree 4

degree 8

degree 16

degree 32

degree 2

degree 4

degree 8

degree 16

degree 32

Figure C.2: The response functions for venusian geoid (top) and topography (middle forsurface and bottom for CMB) at degrees j=2,4,8,16,32. The depth of the mantle is set to theassumed 3,000 km and the viscosity is handled as a constant. For this model (100 layers)the load corresponds to the layer thickness. The amplitude of the dynamic topographyat appropriate boundary could be then evaluated analytically as −h/∆ρ, where h is theparametric layer thickness and ∆ρ stands for the density contrast (at upper boundary 2,900and at CMB 5,500 kg·m−3)


C.3 Inverse Problem

As we have a definition of the forward problem we can model the dynamic geoid and to-pography for a given 3D density structure and 1D viscosity profile. Because of the resultsnonlinear dependency on viscosity such an inverse problem could be solved only by appro-priate numerical methods such as the Monte Carlo random search, a genetic algorithm orsimulated annealing method (see review in Tarantola, 1987 or Matas, 1995). Thesetechniques are computationally demanding and could give us a broad spectrum of differingresults.

Nevertheless we can divide the algorithm into the two-step inversion – in each roundof search we first choose the viscosity by some previously mentioned technique and thenderive the density structure which produce the smallest L2 norm difference between theobserved and predicted data. To obtain a unique solution we consider only lateral densityvariations, thus the mantle model should have a plume-like character – this is only a kind ofmathematical representation which should not be confused with the real unknown venusianmantle structure. However, it could be proved (Kiefer et al., 1986) that such a mantleflow model does not have a high sensitivity to the exact density distribution.

For the linear problem in step two we can calculate the analytical solution by the mini-mization of the least-square misfit between the observed and predicted data. This specificmisfit function we define as:

S2(ρ, η) =jmax∑

j=0

j∑

m=−j

S2jm(ρ, η) (C.22)

S2jm(ρ, η) =

(tpredjm − tobs

jm)(tpredjm − tobs

jm)∗

σ2j

+ (gpredjm − gobs

jm)(gpredjm − gobs

jm)∗, (C.23)

where asterisk denotes the complex conjunction, gpredjm and gobs

jm respectively predicted andobserved spherical harmonic coefficient of gravitational potential (the same for topography)and σ2

j is a degree a priori weighting function whose purpose is to balance power signalsfrom topography and the geoid:

σ2j =

j∑m=−j

tobsjmtobs∗

jm

j∑m=−j

gobsjmgobs∗

jm

. (C.24)

Because gpredjm and tpred

jm are linearly dependent on density, we can decompose them intoa density term ρjm (we can omit the r dependency because there are only lateral variances)and response functions gj, tj (neither C.16 nor C.17 are dependant on m) which are depthintegrals of a full-mantle load (unit mass and given viscosity structure) response functions:

tpredjm = ρjmtj(η) (C.25)

gpredjm = ρjmgj(η), (C.26)


where the best ρjm for the given viscosity profile is uniquely given by the derivative of amisfit function equal to zero – then we can analytically find the solution:

∂S2jm

∂ρjm

= 0 (C.27)

ρjm =

tjtobsjm

σ2j

+ gjgobsjm

tjtjσ2

j+ gjgj

. (C.28)

When we look at the case where the topography is of purely dynamic origin (100% dy-namic support) we can use only the equation C.25 and then the resulting term for densitythen becomes very simple. However we do not consider these modifications because the ob-tained gravitational potential has low both spectral and spatial correlation with the observeddata and moreover solely the dynamic support of the topography for selected broad spec-tral intervals is unrealistic – more desired is to divide the modelling error equally betweenpredicted geoid and topography.

Finally to evaluate the level of overall agreement between the observed and predictedgravitational potential spectra we can examine the correlation (C.29) or percentage of fitteddata quantity (C.30) for every obtained result. These can give us a valuable informationabout quality of our models.

c =

∑jm

gpredjm gobs

jm∗

√∑jm

gpredjm gpred

jm

∗√

∑jm

gobsjmgobs

jm∗

(C.29)

p = 1− ||gpred − gobs||2||gobs||2 (C.30)

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Internet References

[i1] National Space Science Data Centerhttp://nssdc.gsfc.nasa.gov

[i2] University Corporation for Atmospheric Researchhttp://www.windows.ucar.edu

[i3] Massachusetts Institute of Technologyhttp://www.mit.edu

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[i5] Homepage of Don Mitchellhttp://www.mentallandscape.com

[i6] U.S. National Imagery and Mapping Agency EGM96 modelhttp://www.nima.mil/GandG/wgsegm

[i7] NOAA’s National Geophysical Data Centerhttp://www.ngdc.noaa.gov

[i8] Eric Wolfram’s World of Mathhttp://mathworld.wolfram.com

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